Research ArticleVibration and Instability of Rotating Composite Thin-WalledShafts with Internal Damping

Ren Yongsheng Zhang Xingqi Liu Yanghang and Chen Xiulong

College of Mechanical and Electronic Engineering Shandong University of Science amp Technology Qingdao 266590 China

Correspondence should be addressed to Ren Yongsheng rys56sohucom

Received 21 May 2014 Accepted 3 July 2014 Published 17 August 2014

Academic Editor Mohammad Elahinia

Copyright copy 2014 Ren Yongsheng et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The dynamical analysis of a rotating thin-walled composite shaft with internal damping is carried out analyticallyThe equations ofmotion are derived using the thin-walled composite beam theory and the principle of virtual work The internal damping of shaftsis introduced by adopting the multiscale damping analysis method Galerkinrsquos method is used to discretize and solve the governingequations Numerical study shows the effect of design parameters on the natural frequencies critical rotating speeds and instabilitythresholds of shafts

1 Introduction

The rotating composite material shafts are being used asstructural elements in many application areas involving therotating machinery systems This is likely to contribute tothe high strength to weight ratio lower vibration level andlonger service life of compositematerials A significantweightsaving can be achieved by the use of compositematerials Alsoby appropriate design of the composite layup configurationorientation and number of plies the improved performanceof the shaft system can be obtained Furthermore the useof composite would permit the use of longer shafts in thesupercritical range than what is possible with conventionalmetallic shafts In the last few years there exist numerousresearches related to predicting critical speeds and naturalfrequencies of composite shaft Zinberg and Symonds [1]investigated the critical speeds of rotating anisotropic cylin-drical shafts based on an equivalent modulus beam theory(EMBT) and dos Reis et al [2] evaluated the shaft of Zinbergand Symonds [1] by the finite element method Kim andBert [3] adopted the thin- and thick-shell theories of first-order approximation to derive the motion equations of therotating composite thin-walled shafts They used this modelto obtain a closed form solution for a simply supported driveshaft and to analyze the critical speeds of composite shaftsSingh and Gupta [4] developed two composite spinning

shaft models employing EMBT and layerwise beam theory(LBT) respectively It was shown that a discrepancy existsbetween the critical speeds obtained from both models forthe unsymmetric laminated composite shaft Chang et al [5]presented a simple spinning composite shaft model basedon a first-order shear deformable beam theory The finiteelementmethod is used here to find the approximate solutionof the system The model was used to analyze the criticalspeeds frequencies mode shapes and transient response ofa particular composite shaft system Gubran and Gupta [6]presented a modified EMBT model to account for the effectsof a stacking sequence and different coupling mechanismsSong et al [7] used Rehfieldrsquos thin-walled beam theory [8]that presented a composite thin-walled shaft model Theeffects of rotatory inertias axial edge load and boundaryconditions on the natural frequencies and stability of thesystem were investigated Ren et al [9] proposed anothercomposite thin-walled shaftmodel bymeans of the compositethin-walled beam theory an asymptotically correct theoryreferred to as variational asymptotically method (VAM)Theflexible composite shaft is assumed to support on bearingswhich are modeled as springs and dampers and containingof the rigid disks mounted on it The natural frequencies andcritical rotating speeds of the rotating composite shaft withthe variation of the lamination angle ratios of length overradius ratios of radius over thickness and shear deformationare then analyzed

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 123271 10 pageshttpdxdoiorg1011552014123271

2 Shock and Vibration

On the other hand composite material shafts have higherinternal damping than conventionalmetallic shaftsHoweveras it has been shown in previous research [10] that internaldamping in rotating assemblies may lead to whirl instabilitiesin high speed rotors Therefore accurate prediction of effectsof internal damping in composite material rotors is essentialSo far there have been less work related to the stability analy-sis particularly effects of internal damping on the stability ofa rotating composite shaft [4 11ndash14] Singh and Gupta [4 11]introduced discrete viscous damping coefficients to accountfor effect of internal damping In a similar approach dynamicinstability analysis of composite shaft has been performedby Montagnier and Hochard [13] Mazzei and Scott [12]Kim et al [14] In above cases internal damping terms wereincluded simply in equations of motion of rotating shaft nointernal damping modeling has been described Sino et al[15] investigated the stability of an internally damped rotatingcomposite shaft Internal damping was introduced by thecomplex constitutive relation of a viscoelastic compositeThe shaft was modeled by finite element method Howeveronly the mechanical coupling effects induced by symmetricalstacking are taken into account in their mode

In the present work an analytical model applicable to thedynamical analysis of rotating composite thin-walled shaftswith internal damping is proposedThismodel is based on thecomposite thin-walled beam theory referred to as variationalasymptotically method (VAM) by Berdichevsky et al [16]The internal damping of shaft is introduced via themultiscaledamping mechanics [17] The equations of motion of thecomposite shafts are derived by the principle of virtual workGalerkinrsquos method is used then to discretize and solve thegoverning equations The natural frequencies critical rotat-ing speeds and instability thresholds are obtained throughnumerical simulations The effect of the ply angle and aspectratio of cross-section are then assessed The validity of themodel is proved by comparing the results with those inliterature and convergence examination

2 Model of Composite Shaft

Consider the slender thin-walled composite shaft given inFigure 1The length of the shaft is denoted by119871 its thickness isdenoted by ℎ the radius of curvature of themiddle is denotedby 119903 and themaximum cross-sectional dimension is denotedby 119889 It is assumed that 119903 ≪ 119871 ℎ ≪ 119903 and the coordinate 120577 ismeasured along the normal to the middle surface within thelimits minusℎ2 ⩽ 120577 ⩽ ℎ2The shaft rotates about its longitudinal119909-axis at a constant rateΩ

21 The Displacement and Strain The components of thedisplacement along the Cartesian coordinate (119909 119910 119911) areexpressed as follows [16]

1199061(119909 119904) = 119906 (119909) minus 119910 (119904) V1015840 (119909) minus 119911 (119904) 119908

1015840(119909) + 119892 (119904 119909)

1199062(119909 119904) = V (119909) minus 119911 (119904) 120593 (119909)

1199063(119909 119904) = 119908 (119909) + 119910 (119904) 120593 (119909)

(1)

where the primes in (1) denote differentiation with respect to119909

z u3

Ω

120593r

y u2

h(s)

d

L

s 2

x u1 1

120577 120577

Figure 1 Geometry and coordinate systems of composite thin-walled shaft

The tangential and normal displacements V2and V120577can be

expressed as follows

V2= 1199062

119889119910

119889119904+ 1199063

119889119911

119889119904

V120577= 1199062

119889119911

119889119904minus 1199063

119889119910

119889119904

(2)

Based on the expressions shown in (1) and (2) the in-plane strain components can be written in terms of thedisplacement variables as follows

12057411

=1205971199061

120597119909

212057412

=1205971199061

120597119904+

120597V2

120597119909

12057422

=120597V2

120597119904+V120577

119903

(3)

22 Equations ofMotion Theequations ofmotion of the shaftfree vibration can be described by a variational form

int

119897

0

(120575119880119904minus 120575119879119904+ 120575119882119904) 119889119909 = 0 (4)

where 120575119880119904 120575119879119904 and 120575119882

119904are the variation of the strain energy

the kinetic energy and the dissipated energy of the cross-section respectively

The variation of the strain energy of the cross-section canbe expressed as follows

120575119880119904= int119860

120575120576119909119910

T[119876119894119895] 120576119909119910

119889119904 119889120577 (5)

where 120576119909119910

T= 12057411 120574

22212057412 119860 is the cross-sectional area

of the shaft and [119876119894119895] is equivalent off-axis stiffness matrix

The variation of the dissipated energy of the cross-sectioncan be expressed as follows [17]

120575119882119904= int119860

120575120576119909119910

T[120578119894119895] [119876119894119895] 120576119909119910

119889119904 119889120577 (6)

where [120578119894119895] is equivalent off-axis damping matrix

Shock and Vibration 3

Further the variation of the kinetic energy of the cross-section with rotating motion is

120575119879119904= int119860

120575119877T[diag (120588)] 119889119904 119889120577 (7)

where [diag(120588)] is a diagonal matrix with components equalto the mass density 120588 of a ply

The position velocity and acceleration vectors for thedeformed shaft are described as follows

119877 = (119910 + 1199062) i + (119911 + 119906

3) j + (119909 + 119906

1) k

= (2minus Ω (119911 + 119906

3)) i + (

3+ Ω (119910 + 119906

2)) j +

1k

= [2minus 2Ω

3minus Ω2(119910 + 119906

2)] i

+ [3+ 2Ω

2minus Ω2(119911 + 119906

3)] j +

1k

(8)

23 Equivalent Cross-Section Stiffness Matrix For the case ofno internal pressure acting on the shaft (5) can be simplifiedby using free hoop stress resultant (119873

22= 0) assumption as

follows

120575119880119904=

1

2∮ 12057512057411 120575120574

12 [119860 119861

119861 119862]

12057411

12057412

119889119904 (9)

where ∮(sdot)119889119904 denotes the integral around the loop of themidline cross-section and the reduced axial coupling andshear stiffness 119860 119861 and 119862 can be written as follows

119860 (119904) = 11986011

minus1198602

12

11986022

119861 (119904) = 2 [11986016

minus1198601211986026

11986022

]

119862 (119904) = 4 [11986066

minus1198602

26

11986022

]

119860119894119895=

119873

sum

119896=1

119876(119896)

119894119895(119911119896minus 119911119896minus1

) (119894 119895 = 1 2 6)

(10)

In (9) 120575119880119904can also be expressed with respect to 119906 V 119908

and 120593 by combining (1)ndash(3) and (9) Thus one has

120575119880119904= 120575Δ

T[K] Δ (11)

where Δ is 4 times 1 column matrix of kinematic variablesdefined as Δ

T= (1199061015840

1205931015840

11990810158401015840 V10158401015840) and [K] is 4times 4 symmetric

stiffness matrix Its components 119896119894119895are given as follows

11989611

= ∮(119860 minus1198612

119862)119889119904 +

[∮ (119861119862) 119889119904]2

∮ (1119862) 119889119904

11989612

= [

∮ (119861119862) 119889119904

∮ (1119862) 119889119904

]119860119890

11989613

= minus∮(119860 minus1198612

119862)119911119889119904

minus

[∮ (119861119862) 119889119904∮ (119861119862) 119911 119889119904]

∮ (1119862) 119889119904

11989614

= minus∮(119860 minus1198612

119862)119910119889119904

minus

[∮ (119861119862) 119889119904∮ (119861119862) 119910 119889119904]

∮ (1119862) 119889119904

11989622

= [1

∮ (1119862) 119889119904

]1198602

119890

11989623

= minus[

∮ (119861119862) 119911 119889119904

∮ (1119862) 119889119904

]119860119890

11989624

= minus[

∮ (119861119862) 119910 119889119904

∮ (1119862) 119889119904

]119860119890

11989633

= ∮(119860 minus1198612

119862)1199112119889119904 +

[∮ (119861119862) 119911 119889119904]2

∮ (1119862) 119889119904

11989634

= minus∮(119860 minus1198612

119862)119910119911119889119904

minus

[∮ (119861119862) 119910 119889119904∮ (119861119862) 119911 119889119904]

∮ (1119862) 119889119904

11989644

= ∮(119860 minus1198612

119862)1199102119889119904 +

[∮Γ(119861119862) 119910 119889119904]

2

∮ (1119862) 119889119904

119860119890=

1

2∮(119910

119889119911

119889119904minus 119911

119889119910

119889119904) 119889119904

(12)

24 Equivalent Cross-Section DampingMatrix Similar to thederivation of the previous cross-section stiffness formula-tions the variation of the dissipated energy of the cross-section in terms of the strains 120574

11and 12057412can be modeled as

follows

120575119882119904= ∮ 12057411 120574

12 [119860119889

119861119889

119862119889

119863119889

]12057411

12057412

119889119904 (13)

where

119860119889= 11986011988911

+1198602

12

1198602

22

11986011988922

minus11986012

11986022

(11986011988912

+ 11986011988921

)

119861119889= 11986011988916

+ 11986011988961

+ 21198601211986026

1198602

22

11986011988922

minus11986026

11986022

(11986011988912

+ 11986011988921

)

minus11986012

11986022

(11986011988926

+ 11986011988962

)

119862119889= 4 [119860

11988966+

1198602

26

1198602

22

11986011988922

minus11986026

11986022

(11986011988926

+ 11986011988962

)]

4 Shock and Vibration

119860119889119894119895

= int

ℎ2

minusℎ2

120595119894119897119876119897119895119889120577 = 2

1198732

sum

119896=1

120595119896

119894119897119876119896

119897119895(ℎ119896minus ℎ119896minus1

)

(119894 119895 119897 = 1 2 6)

(14)

The variation of the dissipated energy can be alsoexpressed in terms of the kinematic variables as follows

120575119882119904= 120575Δ

T[C] Δ (15)

where [C] is 4 times 4 symmetric damping matrix The formula-tion of its components 119888

119894119895is analogous to stiffness components

119896119894119895as shown in (12) but the terms 119860 119861 and 119862 in (10) should

be replaced by the terms 119860119889 119861119889 and 119862

119889 respectively

25 Equivalent Cross-Section Mass Substituting (1) into (8)and in view of (7) the variation of the kinetic energy of thecross-section can be obtained as follows

120575119879119904= minus (119868

1120575119906 + 119868

2120575V + 119868

3120575119908 + 119868

4120575120593) (16)

where

1198681= 1198871

1198682= 1198871(V minus 2Ω minus Ω

2V) minus 1198872(2Ω + Ω

2) minus 1198873( minus Ω

2120593)

1198683= 1198871( + 2ΩV minus Ω

2119908) + 119887

2( minus Ω

2120593) minus 119887

3(2Ω + Ω

2)

1198684= 1198872( + 2ΩV minus Ω

2119908) minus 119887

3(V minus 2Ω minus Ω

2V)

+ (1198874+ 1198875) ( minus Ω

2120593)

1198871= int119860

120588119889119904119889120577

1198872= int119860

120588119910 119889119904 119889120577

1198873= int119860

120588119911 119889119904 119889120577

1198874= int119860

1205881199102119889119904 119889120577

1198875= int119860

1205881199112119889119904 119889120577

(17)

26 Approximate Solution Method In order to find theapproximate solution of the rotating composite shaft thequantities 119906(119909 119905) V(119909 119905) 119908(119909 119905) and 120593(119909 119905) are assumed inthe form

119906 (119909 119905) =

119873

sum

119895=1

119860119895120572119895(119909) 119890119894120582119905

120593 (119909 119905) =

119873

sum

119895=1

119861119895120579119895(119909) 119890119894120582119905

V (119909 119905) =

119873

sum

119895=1

119862119895120595119895(119909) 119890119894120582119905

119908 (119909 119905) =

119873

sum

119895=1

119863119895120595119895(119909) 119890119894120582119905

(18)

where120572119895(119909) 120579119895(119909) and120595

119895(119909)aremode shape functionswhich

fulfill all the boundary conditions of the composite shaft 120582is complex eigenvalues of the system 119860

119895 119861119895 119862119895 and 119863

119895are

undetermined constants and 119894 = radicminus1Substituting (18) into the governing equations of motion

equations (4)ndash(6) and applying Galerkinrsquos procedure thefollowing governing equations in matrix form can be found

120575119880T(minus1205822[M] + 119894120582 [G] + 119894120582 [C] + [K]) 119880 = 0 (19)

where 119880T=(1198601 1198602 119860

119873 1198611 1198612 119861

119873 1198621 1198622 119862

119873

1198631 1198632 119863

119873) is a constant vector [M] is the mass matrix

[G] is the gyroscopic matrix [C] is the damping matrix and[K] is the stiffness matrix which also includes contributionfrom the centrifugal forces The detailed expressions of thesematrices are as follows

[M] =

[[[

[

minus1198871119867119894119895

0 0 0

0 minus (1198874+ 1198875) 119871119894119895

minus1198872119872119894119895

1198873119872119894119895

0 1198872119876119894119895

1198871119877119894119895

0

0 minus1198873119876119894119895

0 1198871119877119894119895

]]]

]

[G] =

[[[

[

0 0 0 0

0 0 0 0

0 0 0 21198871119877119894119895Ω

0 0 minus21198871119877119894119895Ω 0

]]]

]

[C] =

[[[

[

11988811119864119894119895

11988812119865119894119895

11988813119866119894119895

11988814119866119894119895

11988812119868119894119895

11988822119869119894119895

11988823119870119894119895

11988824119870119894119895

11988813119873119894119895

11988823119874119894119895

11988833119875119894119895

11988834119875119894119895

11988814119873119894119895

11988824119874119894119895

11988834119875119894119895

11988844119875119894119895

]]]

]

[K] =

[[[[

[

11989611119864119894119895

11989612119865119894119895

11989613119866119894119895

11989614119866119894119895

11989612119868119894119895

11989622119869119894119895+ (1198874+ 1198875) 119871119894119895Ω2

11989623119870119894119895

11989624119870119894119895

11989613119873119894119895

11989623119874119894119895

11989633119875119894119895minus 1198871119877119894119895Ω2

11989634119875119894119895

11989614119873119894119895

11989624119874119894119895

11989634119875119894119895

11989644119875119894119895minus 1198871119877119894119895Ω2

]]]]

]

(20)

Shock and Vibration 5

where

119864119894119895= int

119871

0

12057211989412057210158401015840

119895119889119909

119865119894119895= int

119871

0

12057211989412057910158401015840

119895119889119909

119866119894119895= int

119871

0

120572119894120595101584010158401015840

119895119889119909

119867119894119895= int

119871

0

120572119894120572119895119889119909

119868119894119895= int

119871

0

12057911989412057210158401015840

119895119889119909

119869119894119895= int

119871

0

12057911989412057910158401015840

119895119889119909

119870119894119895= int

119871

0

120579119894120595101584010158401015840

119895119889119909

119871119894119895= int

119871

0

120579119894120579119895119889119909

119872119894119895= int

119871

0

120579119894120595119895119889119909

119873119894119895= int

119871

0

120595119894120572101584010158401015840

119895119889119909

119874119894119895= int

119871

0

120595119894120579101584010158401015840

119895119889119909

119875119894119895= int

119871

0

1205951198941205951015840101584010158401015840

119895119889119909

119876119894119895= int

119871

0

120595119894120579119895119889119909

119877119894119895= int

119871

0

120595119894120595119895119889119909

(119894 119895 = 1 119873)

(21)

From (19) one can obtain the following complex eigen-value problem

det minus1205822 [M] + 119894120582 ([G] + [C]) + [K] = 0 (22)

Complex eigenvalue 120582 can be expressed in the form

120582 = 120590 + 119894120596 (23)

The damping natural frequency or whirl frequency of thesystem is the imaginary part 120596 whereas its real part 120590 givesthe decay or growth of the amplitude of vibration A negativevalue of 120590 indicates a stable motion whereas a positive valueindicates an unstable motion growing exponentially in time

Table 1 Mechanical properties of composite material [18]

120588

(kgm3)11986411

(GPa)11986422

(GPa)11986612

(GPa) 12059212

1205781198971

()1205781198972

()1205781198976

()1672 258 87 35 034 065 234 289

Table 2 Modal frequencies of cantilever composite box beam119871119886 = 1436 119886119887 = 5 [0]

16

ModeNatural frequency (Hz)

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 312 313 313 31Second flapping mdash 1902 1902 198First sweeping 1080 1081 1081 110Second sweeping mdash 6838 6838 656First torsional 3783 3784 3784 377Second torsional 11291 11301 11301 1133

Table 3 Modal frequencies of cantilever composite box beam119871119886 = 1436 119886119887 = 5 [90]

16

ModeNatural frequency (Hz)

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 180 181 181 18Second flapping mdash 1136 1136 115First sweeping 567 620 620 65Second sweeping mdash 3921 3921 397First torsional 3783 3784 3784 377Second torsional 11290 11301 11301 1133

3 Numerical Results

The numerical calculations are performed by considering theshaftmade of graphite-epoxywhose elastic characteristics arelisted in Table 1 The shaft has rectangular cross-section offixed geometrical characteristics width 119886 = 032m length119871 = 45952m andwall thickness ℎ = 001016mwhose layupis [120579]16with clamped-free boundary conditions

In order to examine the influence of the number of modeshape functions used in the solution of the equation onthe accuracy of the results the numerical results of naturalfrequency are shown in Tables 2 and 3 and modal dampingin Tables 4 and 5 for an increasing number of mode shapefunctions where 119871 119886 and 119887 are the length width andheight respectively From these tables it can be seen thatto obtain accurate results of the first two natural frequenciesand dampings no more than five mode shape functions arerequired This indicates clearly that the convergence of thepresent model is quite good

A comparison of predictions using the present modelwith those obtained in [18] is also shown in Tables 2 3 4 and5 A perfect agreement of numerical results with those in [18]can be seen

Figure 2 shows the variation of the first two flexural natu-ral frequencies versus rotating speed for various ply angles As

6 Shock and Vibration

Table 4 Modal dampings of cantilever composite box beam 119871119886 =

1436 119886119887 = 5 [0]16

ModeDamping

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 064 065 065 065Second flapping mdash 069 069 067First sweeping 068 068 068 068Second sweeping mdash 085 085 09First torsional 289 289 289 289Second torsional 233 292 292 289

Table 5 Modal dampings of cantilever composite box beam 119871119886 =

1436 119886119887 = 5 [90]16

ModeDamping

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 239 238 238 235Second flapping mdash 237 237 235First sweeping 303 253 253 235Second sweeping mdash 236 236 237First torsional 289 289 289 289Second torsional 282 292 292 289

it can be seen because of the nonsymmetry of the shaft cross-section (119886119887 = 1) the standstill flexural frequencies about thetwo principal axes (flapping and sweeping denote bendingabout the 119910- and 119911-axis resp) are unequal The behaviors ofthe flapping and sweeping bending frequencies versus rotat-ing speed are very different In fact due to the existence of theCoriolis effect the coupling between flapping and sweepingbending is induced the first decreases until it becomes zerowhile the second continues to increase It is observed thatinstead of a rotating speed there is awhole domain of rotatingspeed in which the first flapping frequency does not existIn this domain the flapping frequency becomes imaginaryvalue implying that the shaft becomes unstableWhen the plyangle is decreased in addition to shift of instability domaintowards larger rotating speeds it is also observed that thedomain of instability is enlarged

Figure 3 shows the variation of the first two flexuraldampings versus rotating speed for various ply angles It canbe seen clearly that as the rotating speed is increased thedamping of flapping bending mode decreases and remainsnegative for all rotating speed so the flapping bending modeis stable From the results of Figure 3 it can be also observedthat the dampings corresponding to sweeping bending modeare negative at low rotating speed and increase with increas-ing rotating speed and at certain value of rotating speed thedampings vanish and then become positive Transformationof damping from a negative to a positive value marks theonset of unstable motion The rotating speed correspondingto zero damping is the threshold of instability of the shaftTheenclosed curves located nearby the threshold of instabilityrepresent that the real parts are conjugate Figure 3 also shows

120579 = 0∘

120579 = 45∘

120579 = 90∘

120579 = 0∘

120579 = 45∘

120579 = 90∘

0 50 100 150 200

45

40

35

30

25

20

15

10

5

0

Freq

uenc

y (H

z)

Rotating speed Ω (rpm)

Figure 2 The variation of natural frequencies with the rotatingspeed for various ply angles (119886119887 = 36 first two flexural modes)

0 50 100 150 200

40

20

0

minus80

minus60

minus40

minus20

minus100

Dam

ping

(1s

)

Rotating speed Ω (rpm)

120579 = 0∘

120579 = 45∘

120579 = 90∘

Figure 3 The variation of dampings with the rotating speed forvarious ply angles (119886119887 = 36 first two flexural modes)

that the enclosed curves are shifted toward larger rotatingspeed but the extent of the enclosed curve is increased withthe decrease of the ply angle

Figure 4 shows the variation of the first two extension-twist natural frequencies versus rotating speed for variousply angles As seen in Figure 4 due to the absence of theCoriolis effect the first extension-twist natural frequency (thetwist is dominant) decreases while the second (the extensionis dominant) remains constant at all rotating speeds FromFigure 4 it is seen that the effect of ply angle on the firstextension-twist natural frequency is significant and is quite

Shock and Vibration 7

0 100 200 300 400 500 600 700

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

250

200

150

100

50

0

120579 = 0∘

120579 = 45∘

120579 = 90∘

120579 = 0∘

120579 = 45∘

120579 = 90∘

Figure 4 The variation of natural frequencies with the rotatingspeed for various ply angles (119886119887 = 36 first two extension-twistmodes)

600

400

200

0

minus200

minus400

minus600

minus8000 100 200 300 400 500 600 700

Dam

ping

(1s

)

Rotating speed Ω (rpm)

120579 = 0∘

120579 = 45∘

120579 = 90∘

Figure 5 The variation of dampings with the rotating speed forvarious ply angles (119886119887 = 36 first two extension-twist modes)

different from the case of flexural mode The maximumcritical speed can be reached when the ply angle 120579 = 45

∘Figure 5 shows the variation of the first two extension-

twist dampings versus rotating speed for various ply anglesIt can be observed that the second extension-twist mode isstable whereas the first extension-twist mode is unstable asthe rotating speed increases above certain value The self-excited range is easily identified from the figure by the signof damping It may also be noted that the effect of ply angleon the threshold of instability is similar to that previouslydescribed for the flexural mode

ab = 12

ab = 36

ab = 72

ab = 12

ab = 36

ab = 72

45

35

40

30

25

20

15

10

5

00 50 100 150 200

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

Figure 6 The variation of natural frequencies with the rotatingspeed for various aspect ratios (120579 = 45

∘ first two flexural modes)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus1400 50 100 150 200

ab = 12

ab = 36

ab = 72

Dam

ping

(1s

)

Rotating speed Ω (rpm)

Figure 7 The variation of dampings with the rotating speed forvarious aspect ratios (120579 = 45

∘ first two flexural modes)

Figure 6 shows the variation of the first two flexuralnatural frequencies versus rotating speed for various aspectratios The results show that the critical rotating speedincreases with the decrease of aspect ratios

Figure 7 shows the variation of the first two flexuraldampings versus rotating speed for various aspect ratiosFrom the results it can be seen that the threshold of instabilityincreases as aspect ratio decreases

Figures 8 and 9 present the effect of aspect ratio onthe natural frequency-rotating speed curves and damping-rotating speed curves for the extension-twist mode respec-tivelyThe results show that the decrease of aspect ratio yields

8 Shock and Vibration

140

120

100

80

60

40

20

00 100 200 300 400 500 600 700

ab = 12

ab = 36

ab = 72

ab = 12

ab = 36

ab = 72

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

Figure 8 The variation of natural frequencies with the rotatingspeed for various aspect ratios (120579 = 45

∘ first two extension-twistmodes)

800

600

400

200

0

minus200

minus400

minus600

minus8000 100 200 300 400 500 600 700

Dam

ping

(1s

)

Rotating speed Ω (rpm)

ab = 12

ab = 36

ab = 72

Figure 9 The variation of dampings with the rotating speed forvarious aspect ratios (120579 = 45

∘ first two extension-twist modes)

a significant increase of the critical rotating speed and thethreshold of instability

Figure 10 shows the effect of ply angle on the criticalrotating speed for the flexural mode It can be seen that asthe ply angle increases the critical rotating speeds decreaseand the maximum critical speed is maximum at 120579 = 0

∘Figure 11 shows the effect of ply angle on the threshold of

instability for the flexural mode It is evident that the generaleffect of the ply angle and aspect ratio on the thresholdof instability is similar to that associated with the criticalrotating speeds By comparing Figure 10 with Figure 11 it

80

70

60

50

40

30

20

10

00

10 20 30 40 50 60 70 80 90

Firs

t crit

ical

spee

d (r

pm)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 10The variation of critical speeds with ply angle for variousaspect ratios (flexural mode)

80

70

60

50

40

30

20

10

00

10 20 30 40 50 60 70 80 90

Inst

abili

ty th

resh

old

(rpm

)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 11 The variation of thresholds of instability with ply anglefor various aspect ratios (flexural mode)

may be noted that the threshold of instability is larger thanthe critical rotating speed and the difference between themincreases as aspect ratio decreasesThis implies that the onsetof instability always occurs after the critical rotating speed

Figures 12 and 13 show the variation of the critical rotatingspeed and threshold of instability for the extension-twistmode respectively From these figures it becomes apparentthat the maximum ones occur at 120579 = 45

∘

4 Conclusion

A model was presented for the study of the dynamicalbehavior of rotating thin-walled composite shaft with inter-nal damping The presented model was used to predict

Shock and Vibration 9

600

550

500

450

350

400

300

250

200

0 10 20 30 40 50 60 70 80 90

Firs

t crit

ical

spee

d (r

pm)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 12The variation of critical speeds with ply angle for variousaspect ratios (extension-twist mode)

600

550

500

450

350

400

300

250

200

0 10 20 30 40 50 60 70 80 90

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Inst

abili

ty th

resh

old

(rpm

)

Figure 13 The variation of thresholds of instability with ply anglefor various aspect ratios (extension-twist mode)

the natural frequencies critical rotating speeds and insta-bility thresholds Theoretical solutions of the free vibrationof the shaft were determined by applying Galerkinrsquos methodFrom the present analysis and the numerical results thefollowing main conclusions were drawn

(1) The developed model provides means of predictingthe natural frequencies critical rotating speeds andinstability thresholds of rotating composite thin-walled shafts with internal damping

(2) The ply angle and aspect ratio affect the vibrationaland instability behavior of shaft significantly

(3) There is an obvious increase in the critical rotatingspeeds and instability thresholds as aspect ratio isdecreased

(4) For the flexural mode critical rotating speed andthreshold of instability have their maximum valuesat 120579 = 0

∘ while for the extension-twist mode themaximum ones occur at 120579 = 45

∘(5) The onset of instability always occurs after the critical

rotating speed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded by the National Natural Sci-ence Foundation of China (Grant no 11272190) ShandongProvincial Natural Science Foundation of China (Grant noZR2011EEM031) and Graduate Innovation Project of Shan-dong University of Science ampTechnology of China (Grant noYC130210)

References

[1] H Zinberg and M F Symonds ldquoThe development of anadvanced composite tail rotor drive shaftrdquo in Proceedings of the26th Annual National Forum of the American Helicopter SocietyWashington DC USA June 1970

[2] H L M dos Reis R B Goldman and P H Verstrate ldquoThin-walled laminated composite cylindrical tubes part III criticalspeed analysisrdquo Journal of Composites Technology and Researchvol 9 no 2 pp 58ndash62 1987

[3] C Kim and C W Bert ldquoCritical speed analysis of laminatedcomposite hollow drive shaftsrdquo Composites Engineering vol 3no 7-8 pp 633ndash643 1993

[4] S P Singh and K Gupta ldquoComposite shaft rotordynamic anal-ysis using a layerwise theoryrdquo Journal of Sound and Vibrationvol 191 no 5 pp 739ndash756 1996

[5] M Y Chang J K Chen and C Y Chang ldquoA simple spinninglaminated composite shaft modelrdquo International Journal ofSolids and Structures vol 41 no 3-4 pp 637ndash662 2004

[6] H B H Gubran and K Gupta ldquoThe effect of stacking sequenceand coupling mechanisms on the natural frequencies of com-posite shaftsrdquo Journal of Sound and Vibration vol 282 no 1-2pp 231ndash248 2005

[7] O Song N Jeong and L Librescu ldquoImplication of conservativeand gyroscopic forces on vibration and stability of an elasticallytailored rotating shaft modeled as a composite thin-walledbeamrdquo Journal of the Acoustical Society of America vol 109 no3 pp 972ndash981 2001

[8] L W Rehfield ldquoDesign analysis methodology for compositerotor bladesrdquo inProceedings of the 7thDoDNASAConference onFibrous Composites in Structural Design AFWAL-TR-85-3094pp V(a)1ndashV(a)15 Denver Colo USA 1985

[9] Y S Ren Q Y Dai and X Q Zhang ldquoModeling and dynamicanalysis of rotating composite shaftrdquo Journal of Vibroengineer-ing vol 15 no 4 pp 1816ndash1832 2013

[10] H L Wettergren and K O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous damping sup-ported in anisotropic bearingsrdquo Journal of Sound and Vibrationvol 195 no 1 pp 75ndash84 1996

10 Shock and Vibration

[11] S P Singh and K Gupta ldquoFree damped flexural vibrationanalysis of composite cylindrical tubes using beam and shelltheoriesrdquo Journal of Sound and Vibration vol 172 no 2 pp 171ndash190 1994

[12] A JMazzei andRA Scott ldquoEffects of internal viscous dampingon the stability of a rotating shaft driven through a universaljointrdquo Journal of Sound and Vibration vol 265 no 4 pp 863ndash885 2003

[13] O Montagnier and C Hochard ldquoDynamic instability of super-critical driveshafts mounted on dissipative supports-effects ofviscous and hysteretic internal dampingrdquo Journal of Sound andVibration vol 305 no 3 pp 378ndash400 2007

[14] W Kim A Argento and R A Scott ldquoForced vibration anddynamic stability of a rotating tapered composite Timoshenkoshaft bending motions in end-milling operationsrdquo Journal ofSound and Vibration vol 246 no 4 pp 583ndash600 2001

[15] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquo Composites Science andTechnology vol 68 no 2 pp 337ndash345 2008

[16] V Berdichevsky E Armanios and A Badir ldquoTheory ofanisotropic thin-walled closed-cross-section beamsrdquo Compos-ites Engineering vol 2 no 5ndash7 pp 411ndash432 1992

[17] Y S Ren X H Du S S Sun and X M Teng ldquoStructuraldamping of thin-walled composite one-cell beamsrdquo Journal ofVibration and Shock vol 31 no 3 pp 141ndash152 2012

[18] D A Saravanos D Varelis T S Plagianakos and N Chryso-choidis ldquoA shear beam finite element for the damping analysisof tubular laminated composite beamsrdquo Journal of Sound andVibration vol 291 no 3-5 pp 802ndash823 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

2 Shock and Vibration

On the other hand composite material shafts have higherinternal damping than conventionalmetallic shaftsHoweveras it has been shown in previous research [10] that internaldamping in rotating assemblies may lead to whirl instabilitiesin high speed rotors Therefore accurate prediction of effectsof internal damping in composite material rotors is essentialSo far there have been less work related to the stability analy-sis particularly effects of internal damping on the stability ofa rotating composite shaft [4 11ndash14] Singh and Gupta [4 11]introduced discrete viscous damping coefficients to accountfor effect of internal damping In a similar approach dynamicinstability analysis of composite shaft has been performedby Montagnier and Hochard [13] Mazzei and Scott [12]Kim et al [14] In above cases internal damping terms wereincluded simply in equations of motion of rotating shaft nointernal damping modeling has been described Sino et al[15] investigated the stability of an internally damped rotatingcomposite shaft Internal damping was introduced by thecomplex constitutive relation of a viscoelastic compositeThe shaft was modeled by finite element method Howeveronly the mechanical coupling effects induced by symmetricalstacking are taken into account in their mode

In the present work an analytical model applicable to thedynamical analysis of rotating composite thin-walled shaftswith internal damping is proposedThismodel is based on thecomposite thin-walled beam theory referred to as variationalasymptotically method (VAM) by Berdichevsky et al [16]The internal damping of shaft is introduced via themultiscaledamping mechanics [17] The equations of motion of thecomposite shafts are derived by the principle of virtual workGalerkinrsquos method is used then to discretize and solve thegoverning equations The natural frequencies critical rotat-ing speeds and instability thresholds are obtained throughnumerical simulations The effect of the ply angle and aspectratio of cross-section are then assessed The validity of themodel is proved by comparing the results with those inliterature and convergence examination

2 Model of Composite Shaft

Consider the slender thin-walled composite shaft given inFigure 1The length of the shaft is denoted by119871 its thickness isdenoted by ℎ the radius of curvature of themiddle is denotedby 119903 and themaximum cross-sectional dimension is denotedby 119889 It is assumed that 119903 ≪ 119871 ℎ ≪ 119903 and the coordinate 120577 ismeasured along the normal to the middle surface within thelimits minusℎ2 ⩽ 120577 ⩽ ℎ2The shaft rotates about its longitudinal119909-axis at a constant rateΩ

21 The Displacement and Strain The components of thedisplacement along the Cartesian coordinate (119909 119910 119911) areexpressed as follows [16]

1199061(119909 119904) = 119906 (119909) minus 119910 (119904) V1015840 (119909) minus 119911 (119904) 119908

1015840(119909) + 119892 (119904 119909)

1199062(119909 119904) = V (119909) minus 119911 (119904) 120593 (119909)

1199063(119909 119904) = 119908 (119909) + 119910 (119904) 120593 (119909)

(1)

where the primes in (1) denote differentiation with respect to119909

z u3

Ω

120593r

y u2

h(s)

d

L

s 2

x u1 1

120577 120577

Figure 1 Geometry and coordinate systems of composite thin-walled shaft

The tangential and normal displacements V2and V120577can be

expressed as follows

V2= 1199062

119889119910

119889119904+ 1199063

119889119911

119889119904

V120577= 1199062

119889119911

119889119904minus 1199063

119889119910

119889119904

(2)

Based on the expressions shown in (1) and (2) the in-plane strain components can be written in terms of thedisplacement variables as follows

12057411

=1205971199061

120597119909

212057412

=1205971199061

120597119904+

120597V2

120597119909

12057422

=120597V2

120597119904+V120577

119903

(3)

22 Equations ofMotion Theequations ofmotion of the shaftfree vibration can be described by a variational form

int

119897

0

(120575119880119904minus 120575119879119904+ 120575119882119904) 119889119909 = 0 (4)

where 120575119880119904 120575119879119904 and 120575119882

119904are the variation of the strain energy

the kinetic energy and the dissipated energy of the cross-section respectively

The variation of the strain energy of the cross-section canbe expressed as follows

120575119880119904= int119860

120575120576119909119910

T[119876119894119895] 120576119909119910

119889119904 119889120577 (5)

where 120576119909119910

T= 12057411 120574

22212057412 119860 is the cross-sectional area

of the shaft and [119876119894119895] is equivalent off-axis stiffness matrix

The variation of the dissipated energy of the cross-sectioncan be expressed as follows [17]

120575119882119904= int119860

120575120576119909119910

T[120578119894119895] [119876119894119895] 120576119909119910

119889119904 119889120577 (6)

where [120578119894119895] is equivalent off-axis damping matrix

Shock and Vibration 3

Further the variation of the kinetic energy of the cross-section with rotating motion is

120575119879119904= int119860

120575119877T[diag (120588)] 119889119904 119889120577 (7)

where [diag(120588)] is a diagonal matrix with components equalto the mass density 120588 of a ply

The position velocity and acceleration vectors for thedeformed shaft are described as follows

119877 = (119910 + 1199062) i + (119911 + 119906

3) j + (119909 + 119906

1) k

= (2minus Ω (119911 + 119906

3)) i + (

3+ Ω (119910 + 119906

2)) j +

1k

= [2minus 2Ω

3minus Ω2(119910 + 119906

2)] i

+ [3+ 2Ω

2minus Ω2(119911 + 119906

3)] j +

1k

(8)

23 Equivalent Cross-Section Stiffness Matrix For the case ofno internal pressure acting on the shaft (5) can be simplifiedby using free hoop stress resultant (119873

22= 0) assumption as

follows

120575119880119904=

1

2∮ 12057512057411 120575120574

12 [119860 119861

119861 119862]

12057411

12057412

119889119904 (9)

where ∮(sdot)119889119904 denotes the integral around the loop of themidline cross-section and the reduced axial coupling andshear stiffness 119860 119861 and 119862 can be written as follows

119860 (119904) = 11986011

minus1198602

12

11986022

119861 (119904) = 2 [11986016

minus1198601211986026

11986022

]

119862 (119904) = 4 [11986066

minus1198602

26

11986022

]

119860119894119895=

119873

sum

119896=1

119876(119896)

119894119895(119911119896minus 119911119896minus1

) (119894 119895 = 1 2 6)

(10)

In (9) 120575119880119904can also be expressed with respect to 119906 V 119908

and 120593 by combining (1)ndash(3) and (9) Thus one has

120575119880119904= 120575Δ

T[K] Δ (11)

where Δ is 4 times 1 column matrix of kinematic variablesdefined as Δ

T= (1199061015840

1205931015840

11990810158401015840 V10158401015840) and [K] is 4times 4 symmetric

stiffness matrix Its components 119896119894119895are given as follows

11989611

= ∮(119860 minus1198612

119862)119889119904 +

[∮ (119861119862) 119889119904]2

∮ (1119862) 119889119904

11989612

= [

∮ (119861119862) 119889119904

∮ (1119862) 119889119904

]119860119890

11989613

= minus∮(119860 minus1198612

119862)119911119889119904

minus

[∮ (119861119862) 119889119904∮ (119861119862) 119911 119889119904]

∮ (1119862) 119889119904

11989614

= minus∮(119860 minus1198612

119862)119910119889119904

minus

[∮ (119861119862) 119889119904∮ (119861119862) 119910 119889119904]

∮ (1119862) 119889119904

11989622

= [1

∮ (1119862) 119889119904

]1198602

119890

11989623

= minus[

∮ (119861119862) 119911 119889119904

∮ (1119862) 119889119904

]119860119890

11989624

= minus[

∮ (119861119862) 119910 119889119904

∮ (1119862) 119889119904

]119860119890

11989633

= ∮(119860 minus1198612

119862)1199112119889119904 +

[∮ (119861119862) 119911 119889119904]2

∮ (1119862) 119889119904

11989634

= minus∮(119860 minus1198612

119862)119910119911119889119904

minus

[∮ (119861119862) 119910 119889119904∮ (119861119862) 119911 119889119904]

∮ (1119862) 119889119904

11989644

= ∮(119860 minus1198612

119862)1199102119889119904 +

[∮Γ(119861119862) 119910 119889119904]

2

∮ (1119862) 119889119904

119860119890=

1

2∮(119910

119889119911

119889119904minus 119911

119889119910

119889119904) 119889119904

(12)

24 Equivalent Cross-Section DampingMatrix Similar to thederivation of the previous cross-section stiffness formula-tions the variation of the dissipated energy of the cross-section in terms of the strains 120574

11and 12057412can be modeled as

follows

120575119882119904= ∮ 12057411 120574

12 [119860119889

119861119889

119862119889

119863119889

]12057411

12057412

119889119904 (13)

where

119860119889= 11986011988911

+1198602

12

1198602

22

11986011988922

minus11986012

11986022

(11986011988912

+ 11986011988921

)

119861119889= 11986011988916

+ 11986011988961

+ 21198601211986026

1198602

22

11986011988922

minus11986026

11986022

(11986011988912

+ 11986011988921

)

minus11986012

11986022

(11986011988926

+ 11986011988962

)

119862119889= 4 [119860

11988966+

1198602

26

1198602

22

11986011988922

minus11986026

11986022

(11986011988926

+ 11986011988962

)]

4 Shock and Vibration

119860119889119894119895

= int

ℎ2

minusℎ2

120595119894119897119876119897119895119889120577 = 2

1198732

sum

119896=1

120595119896

119894119897119876119896

119897119895(ℎ119896minus ℎ119896minus1

)

(119894 119895 119897 = 1 2 6)

(14)

The variation of the dissipated energy can be alsoexpressed in terms of the kinematic variables as follows

120575119882119904= 120575Δ

T[C] Δ (15)

where [C] is 4 times 4 symmetric damping matrix The formula-tion of its components 119888

119894119895is analogous to stiffness components

119896119894119895as shown in (12) but the terms 119860 119861 and 119862 in (10) should

be replaced by the terms 119860119889 119861119889 and 119862

119889 respectively

25 Equivalent Cross-Section Mass Substituting (1) into (8)and in view of (7) the variation of the kinetic energy of thecross-section can be obtained as follows

120575119879119904= minus (119868

1120575119906 + 119868

2120575V + 119868

3120575119908 + 119868

4120575120593) (16)

where

1198681= 1198871

1198682= 1198871(V minus 2Ω minus Ω

2V) minus 1198872(2Ω + Ω

2) minus 1198873( minus Ω

2120593)

1198683= 1198871( + 2ΩV minus Ω

2119908) + 119887

2( minus Ω

2120593) minus 119887

3(2Ω + Ω

2)

1198684= 1198872( + 2ΩV minus Ω

2119908) minus 119887

3(V minus 2Ω minus Ω

2V)

+ (1198874+ 1198875) ( minus Ω

2120593)

1198871= int119860

120588119889119904119889120577

1198872= int119860

120588119910 119889119904 119889120577

1198873= int119860

120588119911 119889119904 119889120577

1198874= int119860

1205881199102119889119904 119889120577

1198875= int119860

1205881199112119889119904 119889120577

(17)

26 Approximate Solution Method In order to find theapproximate solution of the rotating composite shaft thequantities 119906(119909 119905) V(119909 119905) 119908(119909 119905) and 120593(119909 119905) are assumed inthe form

119906 (119909 119905) =

119873

sum

119895=1

119860119895120572119895(119909) 119890119894120582119905

120593 (119909 119905) =

119873

sum

119895=1

119861119895120579119895(119909) 119890119894120582119905

V (119909 119905) =

119873

sum

119895=1

119862119895120595119895(119909) 119890119894120582119905

119908 (119909 119905) =

119873

sum

119895=1

119863119895120595119895(119909) 119890119894120582119905

(18)

where120572119895(119909) 120579119895(119909) and120595

119895(119909)aremode shape functionswhich

fulfill all the boundary conditions of the composite shaft 120582is complex eigenvalues of the system 119860

119895 119861119895 119862119895 and 119863

119895are

undetermined constants and 119894 = radicminus1Substituting (18) into the governing equations of motion

equations (4)ndash(6) and applying Galerkinrsquos procedure thefollowing governing equations in matrix form can be found

120575119880T(minus1205822[M] + 119894120582 [G] + 119894120582 [C] + [K]) 119880 = 0 (19)

where 119880T=(1198601 1198602 119860

119873 1198611 1198612 119861

119873 1198621 1198622 119862

119873

1198631 1198632 119863

119873) is a constant vector [M] is the mass matrix

[G] is the gyroscopic matrix [C] is the damping matrix and[K] is the stiffness matrix which also includes contributionfrom the centrifugal forces The detailed expressions of thesematrices are as follows

[M] =

[[[

[

minus1198871119867119894119895

0 0 0

0 minus (1198874+ 1198875) 119871119894119895

minus1198872119872119894119895

1198873119872119894119895

0 1198872119876119894119895

1198871119877119894119895

0

0 minus1198873119876119894119895

0 1198871119877119894119895

]]]

]

[G] =

[[[

[

0 0 0 0

0 0 0 0

0 0 0 21198871119877119894119895Ω

0 0 minus21198871119877119894119895Ω 0

]]]

]

[C] =

[[[

[

11988811119864119894119895

11988812119865119894119895

11988813119866119894119895

11988814119866119894119895

11988812119868119894119895

11988822119869119894119895

11988823119870119894119895

11988824119870119894119895

11988813119873119894119895

11988823119874119894119895

11988833119875119894119895

11988834119875119894119895

11988814119873119894119895

11988824119874119894119895

11988834119875119894119895

11988844119875119894119895

]]]

]

[K] =

[[[[

[

11989611119864119894119895

11989612119865119894119895

11989613119866119894119895

11989614119866119894119895

11989612119868119894119895

11989622119869119894119895+ (1198874+ 1198875) 119871119894119895Ω2

11989623119870119894119895

11989624119870119894119895

11989613119873119894119895

11989623119874119894119895

11989633119875119894119895minus 1198871119877119894119895Ω2

11989634119875119894119895

11989614119873119894119895

11989624119874119894119895

11989634119875119894119895

11989644119875119894119895minus 1198871119877119894119895Ω2

]]]]

]

(20)

Shock and Vibration 5

where

119864119894119895= int

119871

0

12057211989412057210158401015840

119895119889119909

119865119894119895= int

119871

0

12057211989412057910158401015840

119895119889119909

119866119894119895= int

119871

0

120572119894120595101584010158401015840

119895119889119909

119867119894119895= int

119871

0

120572119894120572119895119889119909

119868119894119895= int

119871

0

12057911989412057210158401015840

119895119889119909

119869119894119895= int

119871

0

12057911989412057910158401015840

119895119889119909

119870119894119895= int

119871

0

120579119894120595101584010158401015840

119895119889119909

119871119894119895= int

119871

0

120579119894120579119895119889119909

119872119894119895= int

119871

0

120579119894120595119895119889119909

119873119894119895= int

119871

0

120595119894120572101584010158401015840

119895119889119909

119874119894119895= int

119871

0

120595119894120579101584010158401015840

119895119889119909

119875119894119895= int

119871

0

1205951198941205951015840101584010158401015840

119895119889119909

119876119894119895= int

119871

0

120595119894120579119895119889119909

119877119894119895= int

119871

0

120595119894120595119895119889119909

(119894 119895 = 1 119873)

(21)

From (19) one can obtain the following complex eigen-value problem

det minus1205822 [M] + 119894120582 ([G] + [C]) + [K] = 0 (22)

Complex eigenvalue 120582 can be expressed in the form

120582 = 120590 + 119894120596 (23)

The damping natural frequency or whirl frequency of thesystem is the imaginary part 120596 whereas its real part 120590 givesthe decay or growth of the amplitude of vibration A negativevalue of 120590 indicates a stable motion whereas a positive valueindicates an unstable motion growing exponentially in time

Table 1 Mechanical properties of composite material [18]

120588

(kgm3)11986411

(GPa)11986422

(GPa)11986612

(GPa) 12059212

1205781198971

()1205781198972

()1205781198976

()1672 258 87 35 034 065 234 289

Table 2 Modal frequencies of cantilever composite box beam119871119886 = 1436 119886119887 = 5 [0]

16

ModeNatural frequency (Hz)

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 312 313 313 31Second flapping mdash 1902 1902 198First sweeping 1080 1081 1081 110Second sweeping mdash 6838 6838 656First torsional 3783 3784 3784 377Second torsional 11291 11301 11301 1133

Table 3 Modal frequencies of cantilever composite box beam119871119886 = 1436 119886119887 = 5 [90]

16

ModeNatural frequency (Hz)

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 180 181 181 18Second flapping mdash 1136 1136 115First sweeping 567 620 620 65Second sweeping mdash 3921 3921 397First torsional 3783 3784 3784 377Second torsional 11290 11301 11301 1133

3 Numerical Results

The numerical calculations are performed by considering theshaftmade of graphite-epoxywhose elastic characteristics arelisted in Table 1 The shaft has rectangular cross-section offixed geometrical characteristics width 119886 = 032m length119871 = 45952m andwall thickness ℎ = 001016mwhose layupis [120579]16with clamped-free boundary conditions

In order to examine the influence of the number of modeshape functions used in the solution of the equation onthe accuracy of the results the numerical results of naturalfrequency are shown in Tables 2 and 3 and modal dampingin Tables 4 and 5 for an increasing number of mode shapefunctions where 119871 119886 and 119887 are the length width andheight respectively From these tables it can be seen thatto obtain accurate results of the first two natural frequenciesand dampings no more than five mode shape functions arerequired This indicates clearly that the convergence of thepresent model is quite good

A comparison of predictions using the present modelwith those obtained in [18] is also shown in Tables 2 3 4 and5 A perfect agreement of numerical results with those in [18]can be seen

Figure 2 shows the variation of the first two flexural natu-ral frequencies versus rotating speed for various ply angles As

6 Shock and Vibration

Table 4 Modal dampings of cantilever composite box beam 119871119886 =

1436 119886119887 = 5 [0]16

ModeDamping

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 064 065 065 065Second flapping mdash 069 069 067First sweeping 068 068 068 068Second sweeping mdash 085 085 09First torsional 289 289 289 289Second torsional 233 292 292 289

Table 5 Modal dampings of cantilever composite box beam 119871119886 =

1436 119886119887 = 5 [90]16

ModeDamping

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 239 238 238 235Second flapping mdash 237 237 235First sweeping 303 253 253 235Second sweeping mdash 236 236 237First torsional 289 289 289 289Second torsional 282 292 292 289

it can be seen because of the nonsymmetry of the shaft cross-section (119886119887 = 1) the standstill flexural frequencies about thetwo principal axes (flapping and sweeping denote bendingabout the 119910- and 119911-axis resp) are unequal The behaviors ofthe flapping and sweeping bending frequencies versus rotat-ing speed are very different In fact due to the existence of theCoriolis effect the coupling between flapping and sweepingbending is induced the first decreases until it becomes zerowhile the second continues to increase It is observed thatinstead of a rotating speed there is awhole domain of rotatingspeed in which the first flapping frequency does not existIn this domain the flapping frequency becomes imaginaryvalue implying that the shaft becomes unstableWhen the plyangle is decreased in addition to shift of instability domaintowards larger rotating speeds it is also observed that thedomain of instability is enlarged

Figure 3 shows the variation of the first two flexuraldampings versus rotating speed for various ply angles It canbe seen clearly that as the rotating speed is increased thedamping of flapping bending mode decreases and remainsnegative for all rotating speed so the flapping bending modeis stable From the results of Figure 3 it can be also observedthat the dampings corresponding to sweeping bending modeare negative at low rotating speed and increase with increas-ing rotating speed and at certain value of rotating speed thedampings vanish and then become positive Transformationof damping from a negative to a positive value marks theonset of unstable motion The rotating speed correspondingto zero damping is the threshold of instability of the shaftTheenclosed curves located nearby the threshold of instabilityrepresent that the real parts are conjugate Figure 3 also shows

120579 = 0∘

120579 = 45∘

120579 = 90∘

120579 = 0∘

120579 = 45∘

120579 = 90∘

0 50 100 150 200

45

40

35

30

25

20

15

10

5

0

Freq

uenc

y (H

z)

Rotating speed Ω (rpm)

Figure 2 The variation of natural frequencies with the rotatingspeed for various ply angles (119886119887 = 36 first two flexural modes)

0 50 100 150 200

40

20

0

minus80

minus60

minus40

minus20

minus100

Dam

ping

(1s

)

Rotating speed Ω (rpm)

120579 = 0∘

120579 = 45∘

120579 = 90∘

Figure 3 The variation of dampings with the rotating speed forvarious ply angles (119886119887 = 36 first two flexural modes)

that the enclosed curves are shifted toward larger rotatingspeed but the extent of the enclosed curve is increased withthe decrease of the ply angle

Figure 4 shows the variation of the first two extension-twist natural frequencies versus rotating speed for variousply angles As seen in Figure 4 due to the absence of theCoriolis effect the first extension-twist natural frequency (thetwist is dominant) decreases while the second (the extensionis dominant) remains constant at all rotating speeds FromFigure 4 it is seen that the effect of ply angle on the firstextension-twist natural frequency is significant and is quite

Shock and Vibration 7

0 100 200 300 400 500 600 700

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

250

200

150

100

50

0

120579 = 0∘

120579 = 45∘

120579 = 90∘

120579 = 0∘

120579 = 45∘

120579 = 90∘

Figure 4 The variation of natural frequencies with the rotatingspeed for various ply angles (119886119887 = 36 first two extension-twistmodes)

600

400

200

0

minus200

minus400

minus600

minus8000 100 200 300 400 500 600 700

Dam

ping

(1s

)

Rotating speed Ω (rpm)

120579 = 0∘

120579 = 45∘

120579 = 90∘

Figure 5 The variation of dampings with the rotating speed forvarious ply angles (119886119887 = 36 first two extension-twist modes)

different from the case of flexural mode The maximumcritical speed can be reached when the ply angle 120579 = 45

∘Figure 5 shows the variation of the first two extension-

twist dampings versus rotating speed for various ply anglesIt can be observed that the second extension-twist mode isstable whereas the first extension-twist mode is unstable asthe rotating speed increases above certain value The self-excited range is easily identified from the figure by the signof damping It may also be noted that the effect of ply angleon the threshold of instability is similar to that previouslydescribed for the flexural mode

ab = 12

ab = 36

ab = 72

ab = 12

ab = 36

ab = 72

45

35

40

30

25

20

15

10

5

00 50 100 150 200

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

Figure 6 The variation of natural frequencies with the rotatingspeed for various aspect ratios (120579 = 45

∘ first two flexural modes)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus1400 50 100 150 200

ab = 12

ab = 36

ab = 72

Dam

ping

(1s

)

Rotating speed Ω (rpm)

Figure 7 The variation of dampings with the rotating speed forvarious aspect ratios (120579 = 45

∘ first two flexural modes)

Figure 6 shows the variation of the first two flexuralnatural frequencies versus rotating speed for various aspectratios The results show that the critical rotating speedincreases with the decrease of aspect ratios

Figure 7 shows the variation of the first two flexuraldampings versus rotating speed for various aspect ratiosFrom the results it can be seen that the threshold of instabilityincreases as aspect ratio decreases

Figures 8 and 9 present the effect of aspect ratio onthe natural frequency-rotating speed curves and damping-rotating speed curves for the extension-twist mode respec-tivelyThe results show that the decrease of aspect ratio yields

8 Shock and Vibration

140

120

100

80

60

40

20

00 100 200 300 400 500 600 700

ab = 12

ab = 36

ab = 72

ab = 12

ab = 36

ab = 72

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

Figure 8 The variation of natural frequencies with the rotatingspeed for various aspect ratios (120579 = 45

∘ first two extension-twistmodes)

800

600

400

200

0

minus200

minus400

minus600

minus8000 100 200 300 400 500 600 700

Dam

ping

(1s

)

Rotating speed Ω (rpm)

ab = 12

ab = 36

ab = 72

Figure 9 The variation of dampings with the rotating speed forvarious aspect ratios (120579 = 45

∘ first two extension-twist modes)

a significant increase of the critical rotating speed and thethreshold of instability

Figure 10 shows the effect of ply angle on the criticalrotating speed for the flexural mode It can be seen that asthe ply angle increases the critical rotating speeds decreaseand the maximum critical speed is maximum at 120579 = 0

∘Figure 11 shows the effect of ply angle on the threshold of

instability for the flexural mode It is evident that the generaleffect of the ply angle and aspect ratio on the thresholdof instability is similar to that associated with the criticalrotating speeds By comparing Figure 10 with Figure 11 it

80

70

60

50

40

30

20

10

00

10 20 30 40 50 60 70 80 90

Firs

t crit

ical

spee

d (r

pm)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 10The variation of critical speeds with ply angle for variousaspect ratios (flexural mode)

80

70

60

50

40

30

20

10

00

10 20 30 40 50 60 70 80 90

Inst

abili

ty th

resh

old

(rpm

)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 11 The variation of thresholds of instability with ply anglefor various aspect ratios (flexural mode)

may be noted that the threshold of instability is larger thanthe critical rotating speed and the difference between themincreases as aspect ratio decreasesThis implies that the onsetof instability always occurs after the critical rotating speed

Figures 12 and 13 show the variation of the critical rotatingspeed and threshold of instability for the extension-twistmode respectively From these figures it becomes apparentthat the maximum ones occur at 120579 = 45

∘

4 Conclusion

A model was presented for the study of the dynamicalbehavior of rotating thin-walled composite shaft with inter-nal damping The presented model was used to predict

Shock and Vibration 9

600

550

500

450

350

400

300

250

200

0 10 20 30 40 50 60 70 80 90

Firs

t crit

ical

spee

d (r

pm)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 12The variation of critical speeds with ply angle for variousaspect ratios (extension-twist mode)

600

550

500

450

350

400

300

250

200

0 10 20 30 40 50 60 70 80 90

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Inst

abili

ty th

resh

old

(rpm

)

Figure 13 The variation of thresholds of instability with ply anglefor various aspect ratios (extension-twist mode)

the natural frequencies critical rotating speeds and insta-bility thresholds Theoretical solutions of the free vibrationof the shaft were determined by applying Galerkinrsquos methodFrom the present analysis and the numerical results thefollowing main conclusions were drawn

(1) The developed model provides means of predictingthe natural frequencies critical rotating speeds andinstability thresholds of rotating composite thin-walled shafts with internal damping

(2) The ply angle and aspect ratio affect the vibrationaland instability behavior of shaft significantly

(3) There is an obvious increase in the critical rotatingspeeds and instability thresholds as aspect ratio isdecreased

(4) For the flexural mode critical rotating speed andthreshold of instability have their maximum valuesat 120579 = 0

∘ while for the extension-twist mode themaximum ones occur at 120579 = 45

∘(5) The onset of instability always occurs after the critical

rotating speed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded by the National Natural Sci-ence Foundation of China (Grant no 11272190) ShandongProvincial Natural Science Foundation of China (Grant noZR2011EEM031) and Graduate Innovation Project of Shan-dong University of Science ampTechnology of China (Grant noYC130210)

References

[1] H Zinberg and M F Symonds ldquoThe development of anadvanced composite tail rotor drive shaftrdquo in Proceedings of the26th Annual National Forum of the American Helicopter SocietyWashington DC USA June 1970

[2] H L M dos Reis R B Goldman and P H Verstrate ldquoThin-walled laminated composite cylindrical tubes part III criticalspeed analysisrdquo Journal of Composites Technology and Researchvol 9 no 2 pp 58ndash62 1987

[3] C Kim and C W Bert ldquoCritical speed analysis of laminatedcomposite hollow drive shaftsrdquo Composites Engineering vol 3no 7-8 pp 633ndash643 1993

[4] S P Singh and K Gupta ldquoComposite shaft rotordynamic anal-ysis using a layerwise theoryrdquo Journal of Sound and Vibrationvol 191 no 5 pp 739ndash756 1996

[5] M Y Chang J K Chen and C Y Chang ldquoA simple spinninglaminated composite shaft modelrdquo International Journal ofSolids and Structures vol 41 no 3-4 pp 637ndash662 2004

[6] H B H Gubran and K Gupta ldquoThe effect of stacking sequenceand coupling mechanisms on the natural frequencies of com-posite shaftsrdquo Journal of Sound and Vibration vol 282 no 1-2pp 231ndash248 2005

[7] O Song N Jeong and L Librescu ldquoImplication of conservativeand gyroscopic forces on vibration and stability of an elasticallytailored rotating shaft modeled as a composite thin-walledbeamrdquo Journal of the Acoustical Society of America vol 109 no3 pp 972ndash981 2001

[8] L W Rehfield ldquoDesign analysis methodology for compositerotor bladesrdquo inProceedings of the 7thDoDNASAConference onFibrous Composites in Structural Design AFWAL-TR-85-3094pp V(a)1ndashV(a)15 Denver Colo USA 1985

[9] Y S Ren Q Y Dai and X Q Zhang ldquoModeling and dynamicanalysis of rotating composite shaftrdquo Journal of Vibroengineer-ing vol 15 no 4 pp 1816ndash1832 2013

[10] H L Wettergren and K O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous damping sup-ported in anisotropic bearingsrdquo Journal of Sound and Vibrationvol 195 no 1 pp 75ndash84 1996

10 Shock and Vibration

[11] S P Singh and K Gupta ldquoFree damped flexural vibrationanalysis of composite cylindrical tubes using beam and shelltheoriesrdquo Journal of Sound and Vibration vol 172 no 2 pp 171ndash190 1994

[12] A JMazzei andRA Scott ldquoEffects of internal viscous dampingon the stability of a rotating shaft driven through a universaljointrdquo Journal of Sound and Vibration vol 265 no 4 pp 863ndash885 2003

[13] O Montagnier and C Hochard ldquoDynamic instability of super-critical driveshafts mounted on dissipative supports-effects ofviscous and hysteretic internal dampingrdquo Journal of Sound andVibration vol 305 no 3 pp 378ndash400 2007

[14] W Kim A Argento and R A Scott ldquoForced vibration anddynamic stability of a rotating tapered composite Timoshenkoshaft bending motions in end-milling operationsrdquo Journal ofSound and Vibration vol 246 no 4 pp 583ndash600 2001

[15] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquo Composites Science andTechnology vol 68 no 2 pp 337ndash345 2008

[16] V Berdichevsky E Armanios and A Badir ldquoTheory ofanisotropic thin-walled closed-cross-section beamsrdquo Compos-ites Engineering vol 2 no 5ndash7 pp 411ndash432 1992

[17] Y S Ren X H Du S S Sun and X M Teng ldquoStructuraldamping of thin-walled composite one-cell beamsrdquo Journal ofVibration and Shock vol 31 no 3 pp 141ndash152 2012

[18] D A Saravanos D Varelis T S Plagianakos and N Chryso-choidis ldquoA shear beam finite element for the damping analysisof tubular laminated composite beamsrdquo Journal of Sound andVibration vol 291 no 3-5 pp 802ndash823 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 3

Further the variation of the kinetic energy of the cross-section with rotating motion is

120575119879119904= int119860

120575119877T[diag (120588)] 119889119904 119889120577 (7)

where [diag(120588)] is a diagonal matrix with components equalto the mass density 120588 of a ply

The position velocity and acceleration vectors for thedeformed shaft are described as follows

119877 = (119910 + 1199062) i + (119911 + 119906

3) j + (119909 + 119906

1) k

= (2minus Ω (119911 + 119906

3)) i + (

3+ Ω (119910 + 119906

2)) j +

1k

= [2minus 2Ω

3minus Ω2(119910 + 119906

2)] i

+ [3+ 2Ω

2minus Ω2(119911 + 119906

3)] j +

1k

(8)

23 Equivalent Cross-Section Stiffness Matrix For the case ofno internal pressure acting on the shaft (5) can be simplifiedby using free hoop stress resultant (119873

22= 0) assumption as

follows

120575119880119904=

1

2∮ 12057512057411 120575120574

12 [119860 119861

119861 119862]

12057411

12057412

119889119904 (9)

where ∮(sdot)119889119904 denotes the integral around the loop of themidline cross-section and the reduced axial coupling andshear stiffness 119860 119861 and 119862 can be written as follows

119860 (119904) = 11986011

minus1198602

12

11986022

119861 (119904) = 2 [11986016

minus1198601211986026

11986022

]

119862 (119904) = 4 [11986066

minus1198602

26

11986022

]

119860119894119895=

119873

sum

119896=1

119876(119896)

119894119895(119911119896minus 119911119896minus1

) (119894 119895 = 1 2 6)

(10)

In (9) 120575119880119904can also be expressed with respect to 119906 V 119908

and 120593 by combining (1)ndash(3) and (9) Thus one has

120575119880119904= 120575Δ

T[K] Δ (11)

where Δ is 4 times 1 column matrix of kinematic variablesdefined as Δ

T= (1199061015840

1205931015840

11990810158401015840 V10158401015840) and [K] is 4times 4 symmetric

stiffness matrix Its components 119896119894119895are given as follows

11989611

= ∮(119860 minus1198612

119862)119889119904 +

[∮ (119861119862) 119889119904]2

∮ (1119862) 119889119904

11989612

= [

∮ (119861119862) 119889119904

∮ (1119862) 119889119904

]119860119890

11989613

= minus∮(119860 minus1198612

119862)119911119889119904

minus

[∮ (119861119862) 119889119904∮ (119861119862) 119911 119889119904]

∮ (1119862) 119889119904

11989614

= minus∮(119860 minus1198612

119862)119910119889119904

minus

[∮ (119861119862) 119889119904∮ (119861119862) 119910 119889119904]

∮ (1119862) 119889119904

11989622

= [1

∮ (1119862) 119889119904

]1198602

119890

11989623

= minus[

∮ (119861119862) 119911 119889119904

∮ (1119862) 119889119904

]119860119890

11989624

= minus[

∮ (119861119862) 119910 119889119904

∮ (1119862) 119889119904

]119860119890

11989633

= ∮(119860 minus1198612

119862)1199112119889119904 +

[∮ (119861119862) 119911 119889119904]2

∮ (1119862) 119889119904

11989634

= minus∮(119860 minus1198612

119862)119910119911119889119904

minus

[∮ (119861119862) 119910 119889119904∮ (119861119862) 119911 119889119904]

∮ (1119862) 119889119904

11989644

= ∮(119860 minus1198612

119862)1199102119889119904 +

[∮Γ(119861119862) 119910 119889119904]

2

∮ (1119862) 119889119904

119860119890=

1

2∮(119910

119889119911

119889119904minus 119911

119889119910

119889119904) 119889119904

(12)

24 Equivalent Cross-Section DampingMatrix Similar to thederivation of the previous cross-section stiffness formula-tions the variation of the dissipated energy of the cross-section in terms of the strains 120574

11and 12057412can be modeled as

follows

120575119882119904= ∮ 12057411 120574

12 [119860119889

119861119889

119862119889

119863119889

]12057411

12057412

119889119904 (13)

where

119860119889= 11986011988911

+1198602

12

1198602

22

11986011988922

minus11986012

11986022

(11986011988912

+ 11986011988921

)

119861119889= 11986011988916

+ 11986011988961

+ 21198601211986026

1198602

22

11986011988922

minus11986026

11986022

(11986011988912

+ 11986011988921

)

minus11986012

11986022

(11986011988926

+ 11986011988962

)

119862119889= 4 [119860

11988966+

1198602

26

1198602

22

11986011988922

minus11986026

11986022

(11986011988926

+ 11986011988962

)]

4 Shock and Vibration

119860119889119894119895

= int

ℎ2

minusℎ2

120595119894119897119876119897119895119889120577 = 2

1198732

sum

119896=1

120595119896

119894119897119876119896

119897119895(ℎ119896minus ℎ119896minus1

)

(119894 119895 119897 = 1 2 6)

(14)

The variation of the dissipated energy can be alsoexpressed in terms of the kinematic variables as follows

120575119882119904= 120575Δ

T[C] Δ (15)

where [C] is 4 times 4 symmetric damping matrix The formula-tion of its components 119888

119894119895is analogous to stiffness components

119896119894119895as shown in (12) but the terms 119860 119861 and 119862 in (10) should

be replaced by the terms 119860119889 119861119889 and 119862

119889 respectively

25 Equivalent Cross-Section Mass Substituting (1) into (8)and in view of (7) the variation of the kinetic energy of thecross-section can be obtained as follows

120575119879119904= minus (119868

1120575119906 + 119868

2120575V + 119868

3120575119908 + 119868

4120575120593) (16)

where

1198681= 1198871

1198682= 1198871(V minus 2Ω minus Ω

2V) minus 1198872(2Ω + Ω

2) minus 1198873( minus Ω

2120593)

1198683= 1198871( + 2ΩV minus Ω

2119908) + 119887

2( minus Ω

2120593) minus 119887

3(2Ω + Ω

2)

1198684= 1198872( + 2ΩV minus Ω

2119908) minus 119887

3(V minus 2Ω minus Ω

2V)

+ (1198874+ 1198875) ( minus Ω

2120593)

1198871= int119860

120588119889119904119889120577

1198872= int119860

120588119910 119889119904 119889120577

1198873= int119860

120588119911 119889119904 119889120577

1198874= int119860

1205881199102119889119904 119889120577

1198875= int119860

1205881199112119889119904 119889120577

(17)

26 Approximate Solution Method In order to find theapproximate solution of the rotating composite shaft thequantities 119906(119909 119905) V(119909 119905) 119908(119909 119905) and 120593(119909 119905) are assumed inthe form

119906 (119909 119905) =

119873

sum

119895=1

119860119895120572119895(119909) 119890119894120582119905

120593 (119909 119905) =

119873

sum

119895=1

119861119895120579119895(119909) 119890119894120582119905

V (119909 119905) =

119873

sum

119895=1

119862119895120595119895(119909) 119890119894120582119905

119908 (119909 119905) =

119873

sum

119895=1

119863119895120595119895(119909) 119890119894120582119905

(18)

where120572119895(119909) 120579119895(119909) and120595

119895(119909)aremode shape functionswhich

fulfill all the boundary conditions of the composite shaft 120582is complex eigenvalues of the system 119860

119895 119861119895 119862119895 and 119863

119895are

undetermined constants and 119894 = radicminus1Substituting (18) into the governing equations of motion

equations (4)ndash(6) and applying Galerkinrsquos procedure thefollowing governing equations in matrix form can be found

120575119880T(minus1205822[M] + 119894120582 [G] + 119894120582 [C] + [K]) 119880 = 0 (19)

where 119880T=(1198601 1198602 119860

119873 1198611 1198612 119861

119873 1198621 1198622 119862

119873

1198631 1198632 119863

119873) is a constant vector [M] is the mass matrix

[G] is the gyroscopic matrix [C] is the damping matrix and[K] is the stiffness matrix which also includes contributionfrom the centrifugal forces The detailed expressions of thesematrices are as follows

[M] =

[[[

[

minus1198871119867119894119895

0 0 0

0 minus (1198874+ 1198875) 119871119894119895

minus1198872119872119894119895

1198873119872119894119895

0 1198872119876119894119895

1198871119877119894119895

0

0 minus1198873119876119894119895

0 1198871119877119894119895

]]]

]

[G] =

[[[

[

0 0 0 0

0 0 0 0

0 0 0 21198871119877119894119895Ω

0 0 minus21198871119877119894119895Ω 0

]]]

]

[C] =

[[[

[

11988811119864119894119895

11988812119865119894119895

11988813119866119894119895

11988814119866119894119895

11988812119868119894119895

11988822119869119894119895

11988823119870119894119895

11988824119870119894119895

11988813119873119894119895

11988823119874119894119895

11988833119875119894119895

11988834119875119894119895

11988814119873119894119895

11988824119874119894119895

11988834119875119894119895

11988844119875119894119895

]]]

]

[K] =

[[[[

[

11989611119864119894119895

11989612119865119894119895

11989613119866119894119895

11989614119866119894119895

11989612119868119894119895

11989622119869119894119895+ (1198874+ 1198875) 119871119894119895Ω2

11989623119870119894119895

11989624119870119894119895

11989613119873119894119895

11989623119874119894119895

11989633119875119894119895minus 1198871119877119894119895Ω2

11989634119875119894119895

11989614119873119894119895

11989624119874119894119895

11989634119875119894119895

11989644119875119894119895minus 1198871119877119894119895Ω2

]]]]

]

(20)

Shock and Vibration 5

where

119864119894119895= int

119871

0

12057211989412057210158401015840

119895119889119909

119865119894119895= int

119871

0

12057211989412057910158401015840

119895119889119909

119866119894119895= int

119871

0

120572119894120595101584010158401015840

119895119889119909

119867119894119895= int

119871

0

120572119894120572119895119889119909

119868119894119895= int

119871

0

12057911989412057210158401015840

119895119889119909

119869119894119895= int

119871

0

12057911989412057910158401015840

119895119889119909

119870119894119895= int

119871

0

120579119894120595101584010158401015840

119895119889119909

119871119894119895= int

119871

0

120579119894120579119895119889119909

119872119894119895= int

119871

0

120579119894120595119895119889119909

119873119894119895= int

119871

0

120595119894120572101584010158401015840

119895119889119909

119874119894119895= int

119871

0

120595119894120579101584010158401015840

119895119889119909

119875119894119895= int

119871

0

1205951198941205951015840101584010158401015840

119895119889119909

119876119894119895= int

119871

0

120595119894120579119895119889119909

119877119894119895= int

119871

0

120595119894120595119895119889119909

(119894 119895 = 1 119873)

(21)

From (19) one can obtain the following complex eigen-value problem

det minus1205822 [M] + 119894120582 ([G] + [C]) + [K] = 0 (22)

Complex eigenvalue 120582 can be expressed in the form

120582 = 120590 + 119894120596 (23)

The damping natural frequency or whirl frequency of thesystem is the imaginary part 120596 whereas its real part 120590 givesthe decay or growth of the amplitude of vibration A negativevalue of 120590 indicates a stable motion whereas a positive valueindicates an unstable motion growing exponentially in time

Table 1 Mechanical properties of composite material [18]

120588

(kgm3)11986411

(GPa)11986422

(GPa)11986612

(GPa) 12059212

1205781198971

()1205781198972

()1205781198976

()1672 258 87 35 034 065 234 289

Table 2 Modal frequencies of cantilever composite box beam119871119886 = 1436 119886119887 = 5 [0]

16

ModeNatural frequency (Hz)

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 312 313 313 31Second flapping mdash 1902 1902 198First sweeping 1080 1081 1081 110Second sweeping mdash 6838 6838 656First torsional 3783 3784 3784 377Second torsional 11291 11301 11301 1133

Table 3 Modal frequencies of cantilever composite box beam119871119886 = 1436 119886119887 = 5 [90]

16

ModeNatural frequency (Hz)

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 180 181 181 18Second flapping mdash 1136 1136 115First sweeping 567 620 620 65Second sweeping mdash 3921 3921 397First torsional 3783 3784 3784 377Second torsional 11290 11301 11301 1133

3 Numerical Results

The numerical calculations are performed by considering theshaftmade of graphite-epoxywhose elastic characteristics arelisted in Table 1 The shaft has rectangular cross-section offixed geometrical characteristics width 119886 = 032m length119871 = 45952m andwall thickness ℎ = 001016mwhose layupis [120579]16with clamped-free boundary conditions

In order to examine the influence of the number of modeshape functions used in the solution of the equation onthe accuracy of the results the numerical results of naturalfrequency are shown in Tables 2 and 3 and modal dampingin Tables 4 and 5 for an increasing number of mode shapefunctions where 119871 119886 and 119887 are the length width andheight respectively From these tables it can be seen thatto obtain accurate results of the first two natural frequenciesand dampings no more than five mode shape functions arerequired This indicates clearly that the convergence of thepresent model is quite good

A comparison of predictions using the present modelwith those obtained in [18] is also shown in Tables 2 3 4 and5 A perfect agreement of numerical results with those in [18]can be seen

Figure 2 shows the variation of the first two flexural natu-ral frequencies versus rotating speed for various ply angles As

6 Shock and Vibration

Table 4 Modal dampings of cantilever composite box beam 119871119886 =

1436 119886119887 = 5 [0]16

ModeDamping

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 064 065 065 065Second flapping mdash 069 069 067First sweeping 068 068 068 068Second sweeping mdash 085 085 09First torsional 289 289 289 289Second torsional 233 292 292 289

Table 5 Modal dampings of cantilever composite box beam 119871119886 =

1436 119886119887 = 5 [90]16

ModeDamping

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 239 238 238 235Second flapping mdash 237 237 235First sweeping 303 253 253 235Second sweeping mdash 236 236 237First torsional 289 289 289 289Second torsional 282 292 292 289

it can be seen because of the nonsymmetry of the shaft cross-section (119886119887 = 1) the standstill flexural frequencies about thetwo principal axes (flapping and sweeping denote bendingabout the 119910- and 119911-axis resp) are unequal The behaviors ofthe flapping and sweeping bending frequencies versus rotat-ing speed are very different In fact due to the existence of theCoriolis effect the coupling between flapping and sweepingbending is induced the first decreases until it becomes zerowhile the second continues to increase It is observed thatinstead of a rotating speed there is awhole domain of rotatingspeed in which the first flapping frequency does not existIn this domain the flapping frequency becomes imaginaryvalue implying that the shaft becomes unstableWhen the plyangle is decreased in addition to shift of instability domaintowards larger rotating speeds it is also observed that thedomain of instability is enlarged

Figure 3 shows the variation of the first two flexuraldampings versus rotating speed for various ply angles It canbe seen clearly that as the rotating speed is increased thedamping of flapping bending mode decreases and remainsnegative for all rotating speed so the flapping bending modeis stable From the results of Figure 3 it can be also observedthat the dampings corresponding to sweeping bending modeare negative at low rotating speed and increase with increas-ing rotating speed and at certain value of rotating speed thedampings vanish and then become positive Transformationof damping from a negative to a positive value marks theonset of unstable motion The rotating speed correspondingto zero damping is the threshold of instability of the shaftTheenclosed curves located nearby the threshold of instabilityrepresent that the real parts are conjugate Figure 3 also shows

120579 = 0∘

120579 = 45∘

120579 = 90∘

120579 = 0∘

120579 = 45∘

120579 = 90∘

0 50 100 150 200

45

40

35

30

25

20

15

10

5

0

Freq

uenc

y (H

z)

Rotating speed Ω (rpm)

Figure 2 The variation of natural frequencies with the rotatingspeed for various ply angles (119886119887 = 36 first two flexural modes)

0 50 100 150 200

40

20

0

minus80

minus60

minus40

minus20

minus100

Dam

ping

(1s

)

Rotating speed Ω (rpm)

120579 = 0∘

120579 = 45∘

120579 = 90∘

Figure 3 The variation of dampings with the rotating speed forvarious ply angles (119886119887 = 36 first two flexural modes)

that the enclosed curves are shifted toward larger rotatingspeed but the extent of the enclosed curve is increased withthe decrease of the ply angle

Figure 4 shows the variation of the first two extension-twist natural frequencies versus rotating speed for variousply angles As seen in Figure 4 due to the absence of theCoriolis effect the first extension-twist natural frequency (thetwist is dominant) decreases while the second (the extensionis dominant) remains constant at all rotating speeds FromFigure 4 it is seen that the effect of ply angle on the firstextension-twist natural frequency is significant and is quite

Shock and Vibration 7

0 100 200 300 400 500 600 700

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

250

200

150

100

50

0

120579 = 0∘

120579 = 45∘

120579 = 90∘

120579 = 0∘

120579 = 45∘

120579 = 90∘

Figure 4 The variation of natural frequencies with the rotatingspeed for various ply angles (119886119887 = 36 first two extension-twistmodes)

600

400

200

0

minus200

minus400

minus600

minus8000 100 200 300 400 500 600 700

Dam

ping

(1s

)

Rotating speed Ω (rpm)

120579 = 0∘

120579 = 45∘

120579 = 90∘

Figure 5 The variation of dampings with the rotating speed forvarious ply angles (119886119887 = 36 first two extension-twist modes)

different from the case of flexural mode The maximumcritical speed can be reached when the ply angle 120579 = 45

∘Figure 5 shows the variation of the first two extension-

twist dampings versus rotating speed for various ply anglesIt can be observed that the second extension-twist mode isstable whereas the first extension-twist mode is unstable asthe rotating speed increases above certain value The self-excited range is easily identified from the figure by the signof damping It may also be noted that the effect of ply angleon the threshold of instability is similar to that previouslydescribed for the flexural mode

ab = 12

ab = 36

ab = 72

ab = 12

ab = 36

ab = 72

45

35

40

30

25

20

15

10

5

00 50 100 150 200

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

Figure 6 The variation of natural frequencies with the rotatingspeed for various aspect ratios (120579 = 45

∘ first two flexural modes)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus1400 50 100 150 200

ab = 12

ab = 36

ab = 72

Dam

ping

(1s

)

Rotating speed Ω (rpm)

Figure 7 The variation of dampings with the rotating speed forvarious aspect ratios (120579 = 45

∘ first two flexural modes)

Figure 6 shows the variation of the first two flexuralnatural frequencies versus rotating speed for various aspectratios The results show that the critical rotating speedincreases with the decrease of aspect ratios

Figure 7 shows the variation of the first two flexuraldampings versus rotating speed for various aspect ratiosFrom the results it can be seen that the threshold of instabilityincreases as aspect ratio decreases

Figures 8 and 9 present the effect of aspect ratio onthe natural frequency-rotating speed curves and damping-rotating speed curves for the extension-twist mode respec-tivelyThe results show that the decrease of aspect ratio yields

8 Shock and Vibration

140

120

100

80

60

40

20

00 100 200 300 400 500 600 700

ab = 12

ab = 36

ab = 72

ab = 12

ab = 36

ab = 72

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

Figure 8 The variation of natural frequencies with the rotatingspeed for various aspect ratios (120579 = 45

∘ first two extension-twistmodes)

800

600

400

200

0

minus200

minus400

minus600

minus8000 100 200 300 400 500 600 700

Dam

ping

(1s

)

Rotating speed Ω (rpm)

ab = 12

ab = 36

ab = 72

Figure 9 The variation of dampings with the rotating speed forvarious aspect ratios (120579 = 45

∘ first two extension-twist modes)

a significant increase of the critical rotating speed and thethreshold of instability

Figure 10 shows the effect of ply angle on the criticalrotating speed for the flexural mode It can be seen that asthe ply angle increases the critical rotating speeds decreaseand the maximum critical speed is maximum at 120579 = 0

∘Figure 11 shows the effect of ply angle on the threshold of

instability for the flexural mode It is evident that the generaleffect of the ply angle and aspect ratio on the thresholdof instability is similar to that associated with the criticalrotating speeds By comparing Figure 10 with Figure 11 it

80

70

60

50

40

30

20

10

00

10 20 30 40 50 60 70 80 90

Firs

t crit

ical

spee

d (r

pm)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 10The variation of critical speeds with ply angle for variousaspect ratios (flexural mode)

80

70

60

50

40

30

20

10

00

10 20 30 40 50 60 70 80 90

Inst

abili

ty th

resh

old

(rpm

)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 11 The variation of thresholds of instability with ply anglefor various aspect ratios (flexural mode)

may be noted that the threshold of instability is larger thanthe critical rotating speed and the difference between themincreases as aspect ratio decreasesThis implies that the onsetof instability always occurs after the critical rotating speed

Figures 12 and 13 show the variation of the critical rotatingspeed and threshold of instability for the extension-twistmode respectively From these figures it becomes apparentthat the maximum ones occur at 120579 = 45

∘

4 Conclusion

A model was presented for the study of the dynamicalbehavior of rotating thin-walled composite shaft with inter-nal damping The presented model was used to predict

Shock and Vibration 9

600

550

500

450

350

400

300

250

200

0 10 20 30 40 50 60 70 80 90

Firs

t crit

ical

spee

d (r

pm)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 12The variation of critical speeds with ply angle for variousaspect ratios (extension-twist mode)

600

550

500

450

350

400

300

250

200

0 10 20 30 40 50 60 70 80 90

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Inst

abili

ty th

resh

old

(rpm

)

Figure 13 The variation of thresholds of instability with ply anglefor various aspect ratios (extension-twist mode)

the natural frequencies critical rotating speeds and insta-bility thresholds Theoretical solutions of the free vibrationof the shaft were determined by applying Galerkinrsquos methodFrom the present analysis and the numerical results thefollowing main conclusions were drawn

(1) The developed model provides means of predictingthe natural frequencies critical rotating speeds andinstability thresholds of rotating composite thin-walled shafts with internal damping

(2) The ply angle and aspect ratio affect the vibrationaland instability behavior of shaft significantly

(3) There is an obvious increase in the critical rotatingspeeds and instability thresholds as aspect ratio isdecreased

(4) For the flexural mode critical rotating speed andthreshold of instability have their maximum valuesat 120579 = 0

∘ while for the extension-twist mode themaximum ones occur at 120579 = 45

∘(5) The onset of instability always occurs after the critical

rotating speed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded by the National Natural Sci-ence Foundation of China (Grant no 11272190) ShandongProvincial Natural Science Foundation of China (Grant noZR2011EEM031) and Graduate Innovation Project of Shan-dong University of Science ampTechnology of China (Grant noYC130210)

References

[1] H Zinberg and M F Symonds ldquoThe development of anadvanced composite tail rotor drive shaftrdquo in Proceedings of the26th Annual National Forum of the American Helicopter SocietyWashington DC USA June 1970

[2] H L M dos Reis R B Goldman and P H Verstrate ldquoThin-walled laminated composite cylindrical tubes part III criticalspeed analysisrdquo Journal of Composites Technology and Researchvol 9 no 2 pp 58ndash62 1987

[3] C Kim and C W Bert ldquoCritical speed analysis of laminatedcomposite hollow drive shaftsrdquo Composites Engineering vol 3no 7-8 pp 633ndash643 1993

[4] S P Singh and K Gupta ldquoComposite shaft rotordynamic anal-ysis using a layerwise theoryrdquo Journal of Sound and Vibrationvol 191 no 5 pp 739ndash756 1996

[5] M Y Chang J K Chen and C Y Chang ldquoA simple spinninglaminated composite shaft modelrdquo International Journal ofSolids and Structures vol 41 no 3-4 pp 637ndash662 2004

[6] H B H Gubran and K Gupta ldquoThe effect of stacking sequenceand coupling mechanisms on the natural frequencies of com-posite shaftsrdquo Journal of Sound and Vibration vol 282 no 1-2pp 231ndash248 2005

[7] O Song N Jeong and L Librescu ldquoImplication of conservativeand gyroscopic forces on vibration and stability of an elasticallytailored rotating shaft modeled as a composite thin-walledbeamrdquo Journal of the Acoustical Society of America vol 109 no3 pp 972ndash981 2001

[8] L W Rehfield ldquoDesign analysis methodology for compositerotor bladesrdquo inProceedings of the 7thDoDNASAConference onFibrous Composites in Structural Design AFWAL-TR-85-3094pp V(a)1ndashV(a)15 Denver Colo USA 1985

[9] Y S Ren Q Y Dai and X Q Zhang ldquoModeling and dynamicanalysis of rotating composite shaftrdquo Journal of Vibroengineer-ing vol 15 no 4 pp 1816ndash1832 2013

[10] H L Wettergren and K O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous damping sup-ported in anisotropic bearingsrdquo Journal of Sound and Vibrationvol 195 no 1 pp 75ndash84 1996

10 Shock and Vibration

[11] S P Singh and K Gupta ldquoFree damped flexural vibrationanalysis of composite cylindrical tubes using beam and shelltheoriesrdquo Journal of Sound and Vibration vol 172 no 2 pp 171ndash190 1994

[12] A JMazzei andRA Scott ldquoEffects of internal viscous dampingon the stability of a rotating shaft driven through a universaljointrdquo Journal of Sound and Vibration vol 265 no 4 pp 863ndash885 2003

[13] O Montagnier and C Hochard ldquoDynamic instability of super-critical driveshafts mounted on dissipative supports-effects ofviscous and hysteretic internal dampingrdquo Journal of Sound andVibration vol 305 no 3 pp 378ndash400 2007

[14] W Kim A Argento and R A Scott ldquoForced vibration anddynamic stability of a rotating tapered composite Timoshenkoshaft bending motions in end-milling operationsrdquo Journal ofSound and Vibration vol 246 no 4 pp 583ndash600 2001

[15] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquo Composites Science andTechnology vol 68 no 2 pp 337ndash345 2008

[16] V Berdichevsky E Armanios and A Badir ldquoTheory ofanisotropic thin-walled closed-cross-section beamsrdquo Compos-ites Engineering vol 2 no 5ndash7 pp 411ndash432 1992

[17] Y S Ren X H Du S S Sun and X M Teng ldquoStructuraldamping of thin-walled composite one-cell beamsrdquo Journal ofVibration and Shock vol 31 no 3 pp 141ndash152 2012

[18] D A Saravanos D Varelis T S Plagianakos and N Chryso-choidis ldquoA shear beam finite element for the damping analysisof tubular laminated composite beamsrdquo Journal of Sound andVibration vol 291 no 3-5 pp 802ndash823 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 Shock and Vibration

119860119889119894119895

= int

ℎ2

minusℎ2

120595119894119897119876119897119895119889120577 = 2

1198732

sum

119896=1

120595119896

119894119897119876119896

119897119895(ℎ119896minus ℎ119896minus1

)

(119894 119895 119897 = 1 2 6)

(14)

The variation of the dissipated energy can be alsoexpressed in terms of the kinematic variables as follows

120575119882119904= 120575Δ

T[C] Δ (15)

where [C] is 4 times 4 symmetric damping matrix The formula-tion of its components 119888

119894119895is analogous to stiffness components

119896119894119895as shown in (12) but the terms 119860 119861 and 119862 in (10) should

be replaced by the terms 119860119889 119861119889 and 119862

119889 respectively

25 Equivalent Cross-Section Mass Substituting (1) into (8)and in view of (7) the variation of the kinetic energy of thecross-section can be obtained as follows

120575119879119904= minus (119868

1120575119906 + 119868

2120575V + 119868

3120575119908 + 119868

4120575120593) (16)

where

1198681= 1198871

1198682= 1198871(V minus 2Ω minus Ω

2V) minus 1198872(2Ω + Ω

2) minus 1198873( minus Ω

2120593)

1198683= 1198871( + 2ΩV minus Ω

2119908) + 119887

2( minus Ω

2120593) minus 119887

3(2Ω + Ω

2)

1198684= 1198872( + 2ΩV minus Ω

2119908) minus 119887

3(V minus 2Ω minus Ω

2V)

+ (1198874+ 1198875) ( minus Ω

2120593)

1198871= int119860

120588119889119904119889120577

1198872= int119860

120588119910 119889119904 119889120577

1198873= int119860

120588119911 119889119904 119889120577

1198874= int119860

1205881199102119889119904 119889120577

1198875= int119860

1205881199112119889119904 119889120577

(17)

26 Approximate Solution Method In order to find theapproximate solution of the rotating composite shaft thequantities 119906(119909 119905) V(119909 119905) 119908(119909 119905) and 120593(119909 119905) are assumed inthe form

119906 (119909 119905) =

119873

sum

119895=1

119860119895120572119895(119909) 119890119894120582119905

120593 (119909 119905) =

119873

sum

119895=1

119861119895120579119895(119909) 119890119894120582119905

V (119909 119905) =

119873

sum

119895=1

119862119895120595119895(119909) 119890119894120582119905

119908 (119909 119905) =

119873

sum

119895=1

119863119895120595119895(119909) 119890119894120582119905

(18)

where120572119895(119909) 120579119895(119909) and120595

119895(119909)aremode shape functionswhich

fulfill all the boundary conditions of the composite shaft 120582is complex eigenvalues of the system 119860

119895 119861119895 119862119895 and 119863

119895are

undetermined constants and 119894 = radicminus1Substituting (18) into the governing equations of motion

equations (4)ndash(6) and applying Galerkinrsquos procedure thefollowing governing equations in matrix form can be found

120575119880T(minus1205822[M] + 119894120582 [G] + 119894120582 [C] + [K]) 119880 = 0 (19)

where 119880T=(1198601 1198602 119860

119873 1198611 1198612 119861

119873 1198621 1198622 119862

119873

1198631 1198632 119863

119873) is a constant vector [M] is the mass matrix

[G] is the gyroscopic matrix [C] is the damping matrix and[K] is the stiffness matrix which also includes contributionfrom the centrifugal forces The detailed expressions of thesematrices are as follows

[M] =

[[[

[

minus1198871119867119894119895

0 0 0

0 minus (1198874+ 1198875) 119871119894119895

minus1198872119872119894119895

1198873119872119894119895

0 1198872119876119894119895

1198871119877119894119895

0

0 minus1198873119876119894119895

0 1198871119877119894119895

]]]

]

[G] =

[[[

[

0 0 0 0

0 0 0 0

0 0 0 21198871119877119894119895Ω

0 0 minus21198871119877119894119895Ω 0

]]]

]

[C] =

[[[

[

11988811119864119894119895

11988812119865119894119895

11988813119866119894119895

11988814119866119894119895

11988812119868119894119895

11988822119869119894119895

11988823119870119894119895

11988824119870119894119895

11988813119873119894119895

11988823119874119894119895

11988833119875119894119895

11988834119875119894119895

11988814119873119894119895

11988824119874119894119895

11988834119875119894119895

11988844119875119894119895

]]]

]

[K] =

[[[[

[

11989611119864119894119895

11989612119865119894119895

11989613119866119894119895

11989614119866119894119895

11989612119868119894119895

11989622119869119894119895+ (1198874+ 1198875) 119871119894119895Ω2

11989623119870119894119895

11989624119870119894119895

11989613119873119894119895

11989623119874119894119895

11989633119875119894119895minus 1198871119877119894119895Ω2

11989634119875119894119895

11989614119873119894119895

11989624119874119894119895

11989634119875119894119895

11989644119875119894119895minus 1198871119877119894119895Ω2

]]]]

]

(20)

Shock and Vibration 5

where

119864119894119895= int

119871

0

12057211989412057210158401015840

119895119889119909

119865119894119895= int

119871

0

12057211989412057910158401015840

119895119889119909

119866119894119895= int

119871

0

120572119894120595101584010158401015840

119895119889119909

119867119894119895= int

119871

0

120572119894120572119895119889119909

119868119894119895= int

119871

0

12057911989412057210158401015840

119895119889119909

119869119894119895= int

119871

0

12057911989412057910158401015840

119895119889119909

119870119894119895= int

119871

0

120579119894120595101584010158401015840

119895119889119909

119871119894119895= int

119871

0

120579119894120579119895119889119909

119872119894119895= int

119871

0

120579119894120595119895119889119909

119873119894119895= int

119871

0

120595119894120572101584010158401015840

119895119889119909

119874119894119895= int

119871

0

120595119894120579101584010158401015840

119895119889119909

119875119894119895= int

119871

0

1205951198941205951015840101584010158401015840

119895119889119909

119876119894119895= int

119871

0

120595119894120579119895119889119909

119877119894119895= int

119871

0

120595119894120595119895119889119909

(119894 119895 = 1 119873)

(21)

From (19) one can obtain the following complex eigen-value problem

det minus1205822 [M] + 119894120582 ([G] + [C]) + [K] = 0 (22)

Complex eigenvalue 120582 can be expressed in the form

120582 = 120590 + 119894120596 (23)

The damping natural frequency or whirl frequency of thesystem is the imaginary part 120596 whereas its real part 120590 givesthe decay or growth of the amplitude of vibration A negativevalue of 120590 indicates a stable motion whereas a positive valueindicates an unstable motion growing exponentially in time

Table 1 Mechanical properties of composite material [18]

120588

(kgm3)11986411

(GPa)11986422

(GPa)11986612

(GPa) 12059212

1205781198971

()1205781198972

()1205781198976

()1672 258 87 35 034 065 234 289

Table 2 Modal frequencies of cantilever composite box beam119871119886 = 1436 119886119887 = 5 [0]

16

ModeNatural frequency (Hz)

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 312 313 313 31Second flapping mdash 1902 1902 198First sweeping 1080 1081 1081 110Second sweeping mdash 6838 6838 656First torsional 3783 3784 3784 377Second torsional 11291 11301 11301 1133

Table 3 Modal frequencies of cantilever composite box beam119871119886 = 1436 119886119887 = 5 [90]

16

ModeNatural frequency (Hz)

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 180 181 181 18Second flapping mdash 1136 1136 115First sweeping 567 620 620 65Second sweeping mdash 3921 3921 397First torsional 3783 3784 3784 377Second torsional 11290 11301 11301 1133

3 Numerical Results

The numerical calculations are performed by considering theshaftmade of graphite-epoxywhose elastic characteristics arelisted in Table 1 The shaft has rectangular cross-section offixed geometrical characteristics width 119886 = 032m length119871 = 45952m andwall thickness ℎ = 001016mwhose layupis [120579]16with clamped-free boundary conditions

In order to examine the influence of the number of modeshape functions used in the solution of the equation onthe accuracy of the results the numerical results of naturalfrequency are shown in Tables 2 and 3 and modal dampingin Tables 4 and 5 for an increasing number of mode shapefunctions where 119871 119886 and 119887 are the length width andheight respectively From these tables it can be seen thatto obtain accurate results of the first two natural frequenciesand dampings no more than five mode shape functions arerequired This indicates clearly that the convergence of thepresent model is quite good

A comparison of predictions using the present modelwith those obtained in [18] is also shown in Tables 2 3 4 and5 A perfect agreement of numerical results with those in [18]can be seen

Figure 2 shows the variation of the first two flexural natu-ral frequencies versus rotating speed for various ply angles As

6 Shock and Vibration

Table 4 Modal dampings of cantilever composite box beam 119871119886 =

1436 119886119887 = 5 [0]16

ModeDamping

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 064 065 065 065Second flapping mdash 069 069 067First sweeping 068 068 068 068Second sweeping mdash 085 085 09First torsional 289 289 289 289Second torsional 233 292 292 289

Table 5 Modal dampings of cantilever composite box beam 119871119886 =

1436 119886119887 = 5 [90]16

ModeDamping

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 239 238 238 235Second flapping mdash 237 237 235First sweeping 303 253 253 235Second sweeping mdash 236 236 237First torsional 289 289 289 289Second torsional 282 292 292 289

it can be seen because of the nonsymmetry of the shaft cross-section (119886119887 = 1) the standstill flexural frequencies about thetwo principal axes (flapping and sweeping denote bendingabout the 119910- and 119911-axis resp) are unequal The behaviors ofthe flapping and sweeping bending frequencies versus rotat-ing speed are very different In fact due to the existence of theCoriolis effect the coupling between flapping and sweepingbending is induced the first decreases until it becomes zerowhile the second continues to increase It is observed thatinstead of a rotating speed there is awhole domain of rotatingspeed in which the first flapping frequency does not existIn this domain the flapping frequency becomes imaginaryvalue implying that the shaft becomes unstableWhen the plyangle is decreased in addition to shift of instability domaintowards larger rotating speeds it is also observed that thedomain of instability is enlarged

Figure 3 shows the variation of the first two flexuraldampings versus rotating speed for various ply angles It canbe seen clearly that as the rotating speed is increased thedamping of flapping bending mode decreases and remainsnegative for all rotating speed so the flapping bending modeis stable From the results of Figure 3 it can be also observedthat the dampings corresponding to sweeping bending modeare negative at low rotating speed and increase with increas-ing rotating speed and at certain value of rotating speed thedampings vanish and then become positive Transformationof damping from a negative to a positive value marks theonset of unstable motion The rotating speed correspondingto zero damping is the threshold of instability of the shaftTheenclosed curves located nearby the threshold of instabilityrepresent that the real parts are conjugate Figure 3 also shows

120579 = 0∘

120579 = 45∘

120579 = 90∘

120579 = 0∘

120579 = 45∘

120579 = 90∘

0 50 100 150 200

45

40

35

30

25

20

15

10

5

0

Freq

uenc

y (H

z)

Rotating speed Ω (rpm)

Figure 2 The variation of natural frequencies with the rotatingspeed for various ply angles (119886119887 = 36 first two flexural modes)

0 50 100 150 200

40

20

0

minus80

minus60

minus40

minus20

minus100

Dam

ping

(1s

)

Rotating speed Ω (rpm)

120579 = 0∘

120579 = 45∘

120579 = 90∘

Figure 3 The variation of dampings with the rotating speed forvarious ply angles (119886119887 = 36 first two flexural modes)

that the enclosed curves are shifted toward larger rotatingspeed but the extent of the enclosed curve is increased withthe decrease of the ply angle

Figure 4 shows the variation of the first two extension-twist natural frequencies versus rotating speed for variousply angles As seen in Figure 4 due to the absence of theCoriolis effect the first extension-twist natural frequency (thetwist is dominant) decreases while the second (the extensionis dominant) remains constant at all rotating speeds FromFigure 4 it is seen that the effect of ply angle on the firstextension-twist natural frequency is significant and is quite

Shock and Vibration 7

0 100 200 300 400 500 600 700

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

250

200

150

100

50

0

120579 = 0∘

120579 = 45∘

120579 = 90∘

120579 = 0∘

120579 = 45∘

120579 = 90∘

Figure 4 The variation of natural frequencies with the rotatingspeed for various ply angles (119886119887 = 36 first two extension-twistmodes)

600

400

200

0

minus200

minus400

minus600

minus8000 100 200 300 400 500 600 700

Dam

ping

(1s

)

Rotating speed Ω (rpm)

120579 = 0∘

120579 = 45∘

120579 = 90∘

Figure 5 The variation of dampings with the rotating speed forvarious ply angles (119886119887 = 36 first two extension-twist modes)

different from the case of flexural mode The maximumcritical speed can be reached when the ply angle 120579 = 45

∘Figure 5 shows the variation of the first two extension-

twist dampings versus rotating speed for various ply anglesIt can be observed that the second extension-twist mode isstable whereas the first extension-twist mode is unstable asthe rotating speed increases above certain value The self-excited range is easily identified from the figure by the signof damping It may also be noted that the effect of ply angleon the threshold of instability is similar to that previouslydescribed for the flexural mode

ab = 12

ab = 36

ab = 72

ab = 12

ab = 36

ab = 72

45

35

40

30

25

20

15

10

5

00 50 100 150 200

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

Figure 6 The variation of natural frequencies with the rotatingspeed for various aspect ratios (120579 = 45

∘ first two flexural modes)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus1400 50 100 150 200

ab = 12

ab = 36

ab = 72

Dam

ping

(1s

)

Rotating speed Ω (rpm)

Figure 7 The variation of dampings with the rotating speed forvarious aspect ratios (120579 = 45

∘ first two flexural modes)

Figure 6 shows the variation of the first two flexuralnatural frequencies versus rotating speed for various aspectratios The results show that the critical rotating speedincreases with the decrease of aspect ratios

Figure 7 shows the variation of the first two flexuraldampings versus rotating speed for various aspect ratiosFrom the results it can be seen that the threshold of instabilityincreases as aspect ratio decreases

Figures 8 and 9 present the effect of aspect ratio onthe natural frequency-rotating speed curves and damping-rotating speed curves for the extension-twist mode respec-tivelyThe results show that the decrease of aspect ratio yields

8 Shock and Vibration

140

120

100

80

60

40

20

00 100 200 300 400 500 600 700

ab = 12

ab = 36

ab = 72

ab = 12

ab = 36

ab = 72

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

Figure 8 The variation of natural frequencies with the rotatingspeed for various aspect ratios (120579 = 45

∘ first two extension-twistmodes)

800

600

400

200

0

minus200

minus400

minus600

minus8000 100 200 300 400 500 600 700

Dam

ping

(1s

)

Rotating speed Ω (rpm)

ab = 12

ab = 36

ab = 72

Figure 9 The variation of dampings with the rotating speed forvarious aspect ratios (120579 = 45

∘ first two extension-twist modes)

a significant increase of the critical rotating speed and thethreshold of instability

Figure 10 shows the effect of ply angle on the criticalrotating speed for the flexural mode It can be seen that asthe ply angle increases the critical rotating speeds decreaseand the maximum critical speed is maximum at 120579 = 0

∘Figure 11 shows the effect of ply angle on the threshold of

instability for the flexural mode It is evident that the generaleffect of the ply angle and aspect ratio on the thresholdof instability is similar to that associated with the criticalrotating speeds By comparing Figure 10 with Figure 11 it

80

70

60

50

40

30

20

10

00

10 20 30 40 50 60 70 80 90

Firs

t crit

ical

spee

d (r

pm)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 10The variation of critical speeds with ply angle for variousaspect ratios (flexural mode)

80

70

60

50

40

30

20

10

00

10 20 30 40 50 60 70 80 90

Inst

abili

ty th

resh

old

(rpm

)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 11 The variation of thresholds of instability with ply anglefor various aspect ratios (flexural mode)

may be noted that the threshold of instability is larger thanthe critical rotating speed and the difference between themincreases as aspect ratio decreasesThis implies that the onsetof instability always occurs after the critical rotating speed

Figures 12 and 13 show the variation of the critical rotatingspeed and threshold of instability for the extension-twistmode respectively From these figures it becomes apparentthat the maximum ones occur at 120579 = 45

∘

4 Conclusion

A model was presented for the study of the dynamicalbehavior of rotating thin-walled composite shaft with inter-nal damping The presented model was used to predict

Shock and Vibration 9

600

550

500

450

350

400

300

250

200

0 10 20 30 40 50 60 70 80 90

Firs

t crit

ical

spee

d (r

pm)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 12The variation of critical speeds with ply angle for variousaspect ratios (extension-twist mode)

600

550

500

450

350

400

300

250

200

0 10 20 30 40 50 60 70 80 90

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Inst

abili

ty th

resh

old

(rpm

)

Figure 13 The variation of thresholds of instability with ply anglefor various aspect ratios (extension-twist mode)

the natural frequencies critical rotating speeds and insta-bility thresholds Theoretical solutions of the free vibrationof the shaft were determined by applying Galerkinrsquos methodFrom the present analysis and the numerical results thefollowing main conclusions were drawn

(1) The developed model provides means of predictingthe natural frequencies critical rotating speeds andinstability thresholds of rotating composite thin-walled shafts with internal damping

(2) The ply angle and aspect ratio affect the vibrationaland instability behavior of shaft significantly

(3) There is an obvious increase in the critical rotatingspeeds and instability thresholds as aspect ratio isdecreased

(4) For the flexural mode critical rotating speed andthreshold of instability have their maximum valuesat 120579 = 0

∘ while for the extension-twist mode themaximum ones occur at 120579 = 45

∘(5) The onset of instability always occurs after the critical

rotating speed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded by the National Natural Sci-ence Foundation of China (Grant no 11272190) ShandongProvincial Natural Science Foundation of China (Grant noZR2011EEM031) and Graduate Innovation Project of Shan-dong University of Science ampTechnology of China (Grant noYC130210)

References

[1] H Zinberg and M F Symonds ldquoThe development of anadvanced composite tail rotor drive shaftrdquo in Proceedings of the26th Annual National Forum of the American Helicopter SocietyWashington DC USA June 1970

[2] H L M dos Reis R B Goldman and P H Verstrate ldquoThin-walled laminated composite cylindrical tubes part III criticalspeed analysisrdquo Journal of Composites Technology and Researchvol 9 no 2 pp 58ndash62 1987

[3] C Kim and C W Bert ldquoCritical speed analysis of laminatedcomposite hollow drive shaftsrdquo Composites Engineering vol 3no 7-8 pp 633ndash643 1993

[4] S P Singh and K Gupta ldquoComposite shaft rotordynamic anal-ysis using a layerwise theoryrdquo Journal of Sound and Vibrationvol 191 no 5 pp 739ndash756 1996

[5] M Y Chang J K Chen and C Y Chang ldquoA simple spinninglaminated composite shaft modelrdquo International Journal ofSolids and Structures vol 41 no 3-4 pp 637ndash662 2004

[6] H B H Gubran and K Gupta ldquoThe effect of stacking sequenceand coupling mechanisms on the natural frequencies of com-posite shaftsrdquo Journal of Sound and Vibration vol 282 no 1-2pp 231ndash248 2005

[7] O Song N Jeong and L Librescu ldquoImplication of conservativeand gyroscopic forces on vibration and stability of an elasticallytailored rotating shaft modeled as a composite thin-walledbeamrdquo Journal of the Acoustical Society of America vol 109 no3 pp 972ndash981 2001

[8] L W Rehfield ldquoDesign analysis methodology for compositerotor bladesrdquo inProceedings of the 7thDoDNASAConference onFibrous Composites in Structural Design AFWAL-TR-85-3094pp V(a)1ndashV(a)15 Denver Colo USA 1985

[9] Y S Ren Q Y Dai and X Q Zhang ldquoModeling and dynamicanalysis of rotating composite shaftrdquo Journal of Vibroengineer-ing vol 15 no 4 pp 1816ndash1832 2013

[10] H L Wettergren and K O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous damping sup-ported in anisotropic bearingsrdquo Journal of Sound and Vibrationvol 195 no 1 pp 75ndash84 1996

10 Shock and Vibration

[11] S P Singh and K Gupta ldquoFree damped flexural vibrationanalysis of composite cylindrical tubes using beam and shelltheoriesrdquo Journal of Sound and Vibration vol 172 no 2 pp 171ndash190 1994

[12] A JMazzei andRA Scott ldquoEffects of internal viscous dampingon the stability of a rotating shaft driven through a universaljointrdquo Journal of Sound and Vibration vol 265 no 4 pp 863ndash885 2003

[13] O Montagnier and C Hochard ldquoDynamic instability of super-critical driveshafts mounted on dissipative supports-effects ofviscous and hysteretic internal dampingrdquo Journal of Sound andVibration vol 305 no 3 pp 378ndash400 2007

[14] W Kim A Argento and R A Scott ldquoForced vibration anddynamic stability of a rotating tapered composite Timoshenkoshaft bending motions in end-milling operationsrdquo Journal ofSound and Vibration vol 246 no 4 pp 583ndash600 2001

[15] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquo Composites Science andTechnology vol 68 no 2 pp 337ndash345 2008

[16] V Berdichevsky E Armanios and A Badir ldquoTheory ofanisotropic thin-walled closed-cross-section beamsrdquo Compos-ites Engineering vol 2 no 5ndash7 pp 411ndash432 1992

[17] Y S Ren X H Du S S Sun and X M Teng ldquoStructuraldamping of thin-walled composite one-cell beamsrdquo Journal ofVibration and Shock vol 31 no 3 pp 141ndash152 2012

[18] D A Saravanos D Varelis T S Plagianakos and N Chryso-choidis ldquoA shear beam finite element for the damping analysisof tubular laminated composite beamsrdquo Journal of Sound andVibration vol 291 no 3-5 pp 802ndash823 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 5

where

119864119894119895= int

119871

0

12057211989412057210158401015840

119895119889119909

119865119894119895= int

119871

0

12057211989412057910158401015840

119895119889119909

119866119894119895= int

119871

0

120572119894120595101584010158401015840

119895119889119909

119867119894119895= int

119871

0

120572119894120572119895119889119909

119868119894119895= int

119871

0

12057911989412057210158401015840

119895119889119909

119869119894119895= int

119871

0

12057911989412057910158401015840

119895119889119909

119870119894119895= int

119871

0

120579119894120595101584010158401015840

119895119889119909

119871119894119895= int

119871

0

120579119894120579119895119889119909

119872119894119895= int

119871

0

120579119894120595119895119889119909

119873119894119895= int

119871

0

120595119894120572101584010158401015840

119895119889119909

119874119894119895= int

119871

0

120595119894120579101584010158401015840

119895119889119909

119875119894119895= int

119871

0

1205951198941205951015840101584010158401015840

119895119889119909

119876119894119895= int

119871

0

120595119894120579119895119889119909

119877119894119895= int

119871

0

120595119894120595119895119889119909

(119894 119895 = 1 119873)

(21)

From (19) one can obtain the following complex eigen-value problem

det minus1205822 [M] + 119894120582 ([G] + [C]) + [K] = 0 (22)

Complex eigenvalue 120582 can be expressed in the form

120582 = 120590 + 119894120596 (23)

The damping natural frequency or whirl frequency of thesystem is the imaginary part 120596 whereas its real part 120590 givesthe decay or growth of the amplitude of vibration A negativevalue of 120590 indicates a stable motion whereas a positive valueindicates an unstable motion growing exponentially in time

Table 1 Mechanical properties of composite material [18]

120588

(kgm3)11986411

(GPa)11986422

(GPa)11986612

(GPa) 12059212

1205781198971

()1205781198972

()1205781198976

()1672 258 87 35 034 065 234 289

Table 2 Modal frequencies of cantilever composite box beam119871119886 = 1436 119886119887 = 5 [0]

16

ModeNatural frequency (Hz)

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 312 313 313 31Second flapping mdash 1902 1902 198First sweeping 1080 1081 1081 110Second sweeping mdash 6838 6838 656First torsional 3783 3784 3784 377Second torsional 11291 11301 11301 1133

Table 3 Modal frequencies of cantilever composite box beam119871119886 = 1436 119886119887 = 5 [90]

16

ModeNatural frequency (Hz)

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 180 181 181 18Second flapping mdash 1136 1136 115First sweeping 567 620 620 65Second sweeping mdash 3921 3921 397First torsional 3783 3784 3784 377Second torsional 11290 11301 11301 1133

3 Numerical Results

The numerical calculations are performed by considering theshaftmade of graphite-epoxywhose elastic characteristics arelisted in Table 1 The shaft has rectangular cross-section offixed geometrical characteristics width 119886 = 032m length119871 = 45952m andwall thickness ℎ = 001016mwhose layupis [120579]16with clamped-free boundary conditions

In order to examine the influence of the number of modeshape functions used in the solution of the equation onthe accuracy of the results the numerical results of naturalfrequency are shown in Tables 2 and 3 and modal dampingin Tables 4 and 5 for an increasing number of mode shapefunctions where 119871 119886 and 119887 are the length width andheight respectively From these tables it can be seen thatto obtain accurate results of the first two natural frequenciesand dampings no more than five mode shape functions arerequired This indicates clearly that the convergence of thepresent model is quite good

A comparison of predictions using the present modelwith those obtained in [18] is also shown in Tables 2 3 4 and5 A perfect agreement of numerical results with those in [18]can be seen

Figure 2 shows the variation of the first two flexural natu-ral frequencies versus rotating speed for various ply angles As

6 Shock and Vibration

Table 4 Modal dampings of cantilever composite box beam 119871119886 =

1436 119886119887 = 5 [0]16

ModeDamping

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 064 065 065 065Second flapping mdash 069 069 067First sweeping 068 068 068 068Second sweeping mdash 085 085 09First torsional 289 289 289 289Second torsional 233 292 292 289

Table 5 Modal dampings of cantilever composite box beam 119871119886 =

1436 119886119887 = 5 [90]16

ModeDamping

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 239 238 238 235Second flapping mdash 237 237 235First sweeping 303 253 253 235Second sweeping mdash 236 236 237First torsional 289 289 289 289Second torsional 282 292 292 289

it can be seen because of the nonsymmetry of the shaft cross-section (119886119887 = 1) the standstill flexural frequencies about thetwo principal axes (flapping and sweeping denote bendingabout the 119910- and 119911-axis resp) are unequal The behaviors ofthe flapping and sweeping bending frequencies versus rotat-ing speed are very different In fact due to the existence of theCoriolis effect the coupling between flapping and sweepingbending is induced the first decreases until it becomes zerowhile the second continues to increase It is observed thatinstead of a rotating speed there is awhole domain of rotatingspeed in which the first flapping frequency does not existIn this domain the flapping frequency becomes imaginaryvalue implying that the shaft becomes unstableWhen the plyangle is decreased in addition to shift of instability domaintowards larger rotating speeds it is also observed that thedomain of instability is enlarged

Figure 3 shows the variation of the first two flexuraldampings versus rotating speed for various ply angles It canbe seen clearly that as the rotating speed is increased thedamping of flapping bending mode decreases and remainsnegative for all rotating speed so the flapping bending modeis stable From the results of Figure 3 it can be also observedthat the dampings corresponding to sweeping bending modeare negative at low rotating speed and increase with increas-ing rotating speed and at certain value of rotating speed thedampings vanish and then become positive Transformationof damping from a negative to a positive value marks theonset of unstable motion The rotating speed correspondingto zero damping is the threshold of instability of the shaftTheenclosed curves located nearby the threshold of instabilityrepresent that the real parts are conjugate Figure 3 also shows

120579 = 0∘

120579 = 45∘

120579 = 90∘

120579 = 0∘

120579 = 45∘

120579 = 90∘

0 50 100 150 200

45

40

35

30

25

20

15

10

5

0

Freq

uenc

y (H

z)

Rotating speed Ω (rpm)

Figure 2 The variation of natural frequencies with the rotatingspeed for various ply angles (119886119887 = 36 first two flexural modes)

0 50 100 150 200

40

20

0

minus80

minus60

minus40

minus20

minus100

Dam

ping

(1s

)

Rotating speed Ω (rpm)

120579 = 0∘

120579 = 45∘

120579 = 90∘

Figure 3 The variation of dampings with the rotating speed forvarious ply angles (119886119887 = 36 first two flexural modes)

that the enclosed curves are shifted toward larger rotatingspeed but the extent of the enclosed curve is increased withthe decrease of the ply angle

Figure 4 shows the variation of the first two extension-twist natural frequencies versus rotating speed for variousply angles As seen in Figure 4 due to the absence of theCoriolis effect the first extension-twist natural frequency (thetwist is dominant) decreases while the second (the extensionis dominant) remains constant at all rotating speeds FromFigure 4 it is seen that the effect of ply angle on the firstextension-twist natural frequency is significant and is quite

Shock and Vibration 7

0 100 200 300 400 500 600 700

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

250

200

150

100

50

0

120579 = 0∘

120579 = 45∘

120579 = 90∘

120579 = 0∘

120579 = 45∘

120579 = 90∘

Figure 4 The variation of natural frequencies with the rotatingspeed for various ply angles (119886119887 = 36 first two extension-twistmodes)

600

400

200

0

minus200

minus400

minus600

minus8000 100 200 300 400 500 600 700

Dam

ping

(1s

)

Rotating speed Ω (rpm)

120579 = 0∘

120579 = 45∘

120579 = 90∘

Figure 5 The variation of dampings with the rotating speed forvarious ply angles (119886119887 = 36 first two extension-twist modes)

different from the case of flexural mode The maximumcritical speed can be reached when the ply angle 120579 = 45

∘Figure 5 shows the variation of the first two extension-

twist dampings versus rotating speed for various ply anglesIt can be observed that the second extension-twist mode isstable whereas the first extension-twist mode is unstable asthe rotating speed increases above certain value The self-excited range is easily identified from the figure by the signof damping It may also be noted that the effect of ply angleon the threshold of instability is similar to that previouslydescribed for the flexural mode

ab = 12

ab = 36

ab = 72

ab = 12

ab = 36

ab = 72

45

35

40

30

25

20

15

10

5

00 50 100 150 200

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

Figure 6 The variation of natural frequencies with the rotatingspeed for various aspect ratios (120579 = 45

∘ first two flexural modes)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus1400 50 100 150 200

ab = 12

ab = 36

ab = 72

Dam

ping

(1s

)

Rotating speed Ω (rpm)

Figure 7 The variation of dampings with the rotating speed forvarious aspect ratios (120579 = 45

∘ first two flexural modes)

Figure 6 shows the variation of the first two flexuralnatural frequencies versus rotating speed for various aspectratios The results show that the critical rotating speedincreases with the decrease of aspect ratios

Figure 7 shows the variation of the first two flexuraldampings versus rotating speed for various aspect ratiosFrom the results it can be seen that the threshold of instabilityincreases as aspect ratio decreases

Figures 8 and 9 present the effect of aspect ratio onthe natural frequency-rotating speed curves and damping-rotating speed curves for the extension-twist mode respec-tivelyThe results show that the decrease of aspect ratio yields

8 Shock and Vibration

140

120

100

80

60

40

20

00 100 200 300 400 500 600 700

ab = 12

ab = 36

ab = 72

ab = 12

ab = 36

ab = 72

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

Figure 8 The variation of natural frequencies with the rotatingspeed for various aspect ratios (120579 = 45

∘ first two extension-twistmodes)

800

600

400

200

0

minus200

minus400

minus600

minus8000 100 200 300 400 500 600 700

Dam

ping

(1s

)

Rotating speed Ω (rpm)

ab = 12

ab = 36

ab = 72

Figure 9 The variation of dampings with the rotating speed forvarious aspect ratios (120579 = 45

∘ first two extension-twist modes)

a significant increase of the critical rotating speed and thethreshold of instability

Figure 10 shows the effect of ply angle on the criticalrotating speed for the flexural mode It can be seen that asthe ply angle increases the critical rotating speeds decreaseand the maximum critical speed is maximum at 120579 = 0

∘Figure 11 shows the effect of ply angle on the threshold of

instability for the flexural mode It is evident that the generaleffect of the ply angle and aspect ratio on the thresholdof instability is similar to that associated with the criticalrotating speeds By comparing Figure 10 with Figure 11 it

80

70

60

50

40

30

20

10

00

10 20 30 40 50 60 70 80 90

Firs

t crit

ical

spee

d (r

pm)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 10The variation of critical speeds with ply angle for variousaspect ratios (flexural mode)

80

70

60

50

40

30

20

10

00

10 20 30 40 50 60 70 80 90

Inst

abili

ty th

resh

old

(rpm

)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 11 The variation of thresholds of instability with ply anglefor various aspect ratios (flexural mode)

may be noted that the threshold of instability is larger thanthe critical rotating speed and the difference between themincreases as aspect ratio decreasesThis implies that the onsetof instability always occurs after the critical rotating speed

Figures 12 and 13 show the variation of the critical rotatingspeed and threshold of instability for the extension-twistmode respectively From these figures it becomes apparentthat the maximum ones occur at 120579 = 45

∘

4 Conclusion

A model was presented for the study of the dynamicalbehavior of rotating thin-walled composite shaft with inter-nal damping The presented model was used to predict

Shock and Vibration 9

600

550

500

450

350

400

300

250

200

0 10 20 30 40 50 60 70 80 90

Firs

t crit

ical

spee

d (r

pm)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 12The variation of critical speeds with ply angle for variousaspect ratios (extension-twist mode)

600

550

500

450

350

400

300

250

200

0 10 20 30 40 50 60 70 80 90

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Inst

abili

ty th

resh

old

(rpm

)

Figure 13 The variation of thresholds of instability with ply anglefor various aspect ratios (extension-twist mode)

the natural frequencies critical rotating speeds and insta-bility thresholds Theoretical solutions of the free vibrationof the shaft were determined by applying Galerkinrsquos methodFrom the present analysis and the numerical results thefollowing main conclusions were drawn

(1) The developed model provides means of predictingthe natural frequencies critical rotating speeds andinstability thresholds of rotating composite thin-walled shafts with internal damping

(2) The ply angle and aspect ratio affect the vibrationaland instability behavior of shaft significantly

(3) There is an obvious increase in the critical rotatingspeeds and instability thresholds as aspect ratio isdecreased

(4) For the flexural mode critical rotating speed andthreshold of instability have their maximum valuesat 120579 = 0

∘ while for the extension-twist mode themaximum ones occur at 120579 = 45

∘(5) The onset of instability always occurs after the critical

rotating speed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded by the National Natural Sci-ence Foundation of China (Grant no 11272190) ShandongProvincial Natural Science Foundation of China (Grant noZR2011EEM031) and Graduate Innovation Project of Shan-dong University of Science ampTechnology of China (Grant noYC130210)

References

[1] H Zinberg and M F Symonds ldquoThe development of anadvanced composite tail rotor drive shaftrdquo in Proceedings of the26th Annual National Forum of the American Helicopter SocietyWashington DC USA June 1970

[2] H L M dos Reis R B Goldman and P H Verstrate ldquoThin-walled laminated composite cylindrical tubes part III criticalspeed analysisrdquo Journal of Composites Technology and Researchvol 9 no 2 pp 58ndash62 1987

[3] C Kim and C W Bert ldquoCritical speed analysis of laminatedcomposite hollow drive shaftsrdquo Composites Engineering vol 3no 7-8 pp 633ndash643 1993

[4] S P Singh and K Gupta ldquoComposite shaft rotordynamic anal-ysis using a layerwise theoryrdquo Journal of Sound and Vibrationvol 191 no 5 pp 739ndash756 1996

[5] M Y Chang J K Chen and C Y Chang ldquoA simple spinninglaminated composite shaft modelrdquo International Journal ofSolids and Structures vol 41 no 3-4 pp 637ndash662 2004

[6] H B H Gubran and K Gupta ldquoThe effect of stacking sequenceand coupling mechanisms on the natural frequencies of com-posite shaftsrdquo Journal of Sound and Vibration vol 282 no 1-2pp 231ndash248 2005

[7] O Song N Jeong and L Librescu ldquoImplication of conservativeand gyroscopic forces on vibration and stability of an elasticallytailored rotating shaft modeled as a composite thin-walledbeamrdquo Journal of the Acoustical Society of America vol 109 no3 pp 972ndash981 2001

[8] L W Rehfield ldquoDesign analysis methodology for compositerotor bladesrdquo inProceedings of the 7thDoDNASAConference onFibrous Composites in Structural Design AFWAL-TR-85-3094pp V(a)1ndashV(a)15 Denver Colo USA 1985

[9] Y S Ren Q Y Dai and X Q Zhang ldquoModeling and dynamicanalysis of rotating composite shaftrdquo Journal of Vibroengineer-ing vol 15 no 4 pp 1816ndash1832 2013

[10] H L Wettergren and K O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous damping sup-ported in anisotropic bearingsrdquo Journal of Sound and Vibrationvol 195 no 1 pp 75ndash84 1996

10 Shock and Vibration

[11] S P Singh and K Gupta ldquoFree damped flexural vibrationanalysis of composite cylindrical tubes using beam and shelltheoriesrdquo Journal of Sound and Vibration vol 172 no 2 pp 171ndash190 1994

[12] A JMazzei andRA Scott ldquoEffects of internal viscous dampingon the stability of a rotating shaft driven through a universaljointrdquo Journal of Sound and Vibration vol 265 no 4 pp 863ndash885 2003

[13] O Montagnier and C Hochard ldquoDynamic instability of super-critical driveshafts mounted on dissipative supports-effects ofviscous and hysteretic internal dampingrdquo Journal of Sound andVibration vol 305 no 3 pp 378ndash400 2007

[14] W Kim A Argento and R A Scott ldquoForced vibration anddynamic stability of a rotating tapered composite Timoshenkoshaft bending motions in end-milling operationsrdquo Journal ofSound and Vibration vol 246 no 4 pp 583ndash600 2001

[15] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquo Composites Science andTechnology vol 68 no 2 pp 337ndash345 2008

[16] V Berdichevsky E Armanios and A Badir ldquoTheory ofanisotropic thin-walled closed-cross-section beamsrdquo Compos-ites Engineering vol 2 no 5ndash7 pp 411ndash432 1992

[17] Y S Ren X H Du S S Sun and X M Teng ldquoStructuraldamping of thin-walled composite one-cell beamsrdquo Journal ofVibration and Shock vol 31 no 3 pp 141ndash152 2012

[18] D A Saravanos D Varelis T S Plagianakos and N Chryso-choidis ldquoA shear beam finite element for the damping analysisof tubular laminated composite beamsrdquo Journal of Sound andVibration vol 291 no 3-5 pp 802ndash823 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Shock and Vibration

Table 4 Modal dampings of cantilever composite box beam 119871119886 =

1436 119886119887 = 5 [0]16

ModeDamping

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 064 065 065 065Second flapping mdash 069 069 067First sweeping 068 068 068 068Second sweeping mdash 085 085 09First torsional 289 289 289 289Second torsional 233 292 292 289

Table 5 Modal dampings of cantilever composite box beam 119871119886 =

1436 119886119887 = 5 [90]16

ModeDamping

Present Reference [18]119873 = 1 119873 = 3 119873 = 5

First flapping 239 238 238 235Second flapping mdash 237 237 235First sweeping 303 253 253 235Second sweeping mdash 236 236 237First torsional 289 289 289 289Second torsional 282 292 292 289

it can be seen because of the nonsymmetry of the shaft cross-section (119886119887 = 1) the standstill flexural frequencies about thetwo principal axes (flapping and sweeping denote bendingabout the 119910- and 119911-axis resp) are unequal The behaviors ofthe flapping and sweeping bending frequencies versus rotat-ing speed are very different In fact due to the existence of theCoriolis effect the coupling between flapping and sweepingbending is induced the first decreases until it becomes zerowhile the second continues to increase It is observed thatinstead of a rotating speed there is awhole domain of rotatingspeed in which the first flapping frequency does not existIn this domain the flapping frequency becomes imaginaryvalue implying that the shaft becomes unstableWhen the plyangle is decreased in addition to shift of instability domaintowards larger rotating speeds it is also observed that thedomain of instability is enlarged

Figure 3 shows the variation of the first two flexuraldampings versus rotating speed for various ply angles It canbe seen clearly that as the rotating speed is increased thedamping of flapping bending mode decreases and remainsnegative for all rotating speed so the flapping bending modeis stable From the results of Figure 3 it can be also observedthat the dampings corresponding to sweeping bending modeare negative at low rotating speed and increase with increas-ing rotating speed and at certain value of rotating speed thedampings vanish and then become positive Transformationof damping from a negative to a positive value marks theonset of unstable motion The rotating speed correspondingto zero damping is the threshold of instability of the shaftTheenclosed curves located nearby the threshold of instabilityrepresent that the real parts are conjugate Figure 3 also shows

120579 = 0∘

120579 = 45∘

120579 = 90∘

120579 = 0∘

120579 = 45∘

120579 = 90∘

0 50 100 150 200

45

40

35

30

25

20

15

10

5

0

Freq

uenc

y (H

z)

Rotating speed Ω (rpm)

Figure 2 The variation of natural frequencies with the rotatingspeed for various ply angles (119886119887 = 36 first two flexural modes)

0 50 100 150 200

40

20

0

minus80

minus60

minus40

minus20

minus100

Dam

ping

(1s

)

Rotating speed Ω (rpm)

120579 = 0∘

120579 = 45∘

120579 = 90∘

Figure 3 The variation of dampings with the rotating speed forvarious ply angles (119886119887 = 36 first two flexural modes)

that the enclosed curves are shifted toward larger rotatingspeed but the extent of the enclosed curve is increased withthe decrease of the ply angle

Figure 4 shows the variation of the first two extension-twist natural frequencies versus rotating speed for variousply angles As seen in Figure 4 due to the absence of theCoriolis effect the first extension-twist natural frequency (thetwist is dominant) decreases while the second (the extensionis dominant) remains constant at all rotating speeds FromFigure 4 it is seen that the effect of ply angle on the firstextension-twist natural frequency is significant and is quite

Shock and Vibration 7

0 100 200 300 400 500 600 700

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

250

200

150

100

50

0

120579 = 0∘

120579 = 45∘

120579 = 90∘

120579 = 0∘

120579 = 45∘

120579 = 90∘

Figure 4 The variation of natural frequencies with the rotatingspeed for various ply angles (119886119887 = 36 first two extension-twistmodes)

600

400

200

0

minus200

minus400

minus600

minus8000 100 200 300 400 500 600 700

Dam

ping

(1s

)

Rotating speed Ω (rpm)

120579 = 0∘

120579 = 45∘

120579 = 90∘

Figure 5 The variation of dampings with the rotating speed forvarious ply angles (119886119887 = 36 first two extension-twist modes)

different from the case of flexural mode The maximumcritical speed can be reached when the ply angle 120579 = 45

∘Figure 5 shows the variation of the first two extension-

twist dampings versus rotating speed for various ply anglesIt can be observed that the second extension-twist mode isstable whereas the first extension-twist mode is unstable asthe rotating speed increases above certain value The self-excited range is easily identified from the figure by the signof damping It may also be noted that the effect of ply angleon the threshold of instability is similar to that previouslydescribed for the flexural mode

ab = 12

ab = 36

ab = 72

ab = 12

ab = 36

ab = 72

45

35

40

30

25

20

15

10

5

00 50 100 150 200

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

Figure 6 The variation of natural frequencies with the rotatingspeed for various aspect ratios (120579 = 45

∘ first two flexural modes)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus1400 50 100 150 200

ab = 12

ab = 36

ab = 72

Dam

ping

(1s

)

Rotating speed Ω (rpm)

Figure 7 The variation of dampings with the rotating speed forvarious aspect ratios (120579 = 45

∘ first two flexural modes)

Figure 6 shows the variation of the first two flexuralnatural frequencies versus rotating speed for various aspectratios The results show that the critical rotating speedincreases with the decrease of aspect ratios

Figure 7 shows the variation of the first two flexuraldampings versus rotating speed for various aspect ratiosFrom the results it can be seen that the threshold of instabilityincreases as aspect ratio decreases

Figures 8 and 9 present the effect of aspect ratio onthe natural frequency-rotating speed curves and damping-rotating speed curves for the extension-twist mode respec-tivelyThe results show that the decrease of aspect ratio yields

8 Shock and Vibration

140

120

100

80

60

40

20

00 100 200 300 400 500 600 700

ab = 12

ab = 36

ab = 72

ab = 12

ab = 36

ab = 72

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

Figure 8 The variation of natural frequencies with the rotatingspeed for various aspect ratios (120579 = 45

∘ first two extension-twistmodes)

800

600

400

200

0

minus200

minus400

minus600

minus8000 100 200 300 400 500 600 700

Dam

ping

(1s

)

Rotating speed Ω (rpm)

ab = 12

ab = 36

ab = 72

Figure 9 The variation of dampings with the rotating speed forvarious aspect ratios (120579 = 45

∘ first two extension-twist modes)

a significant increase of the critical rotating speed and thethreshold of instability

Figure 10 shows the effect of ply angle on the criticalrotating speed for the flexural mode It can be seen that asthe ply angle increases the critical rotating speeds decreaseand the maximum critical speed is maximum at 120579 = 0

∘Figure 11 shows the effect of ply angle on the threshold of

instability for the flexural mode It is evident that the generaleffect of the ply angle and aspect ratio on the thresholdof instability is similar to that associated with the criticalrotating speeds By comparing Figure 10 with Figure 11 it

80

70

60

50

40

30

20

10

00

10 20 30 40 50 60 70 80 90

Firs

t crit

ical

spee

d (r

pm)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 10The variation of critical speeds with ply angle for variousaspect ratios (flexural mode)

80

70

60

50

40

30

20

10

00

10 20 30 40 50 60 70 80 90

Inst

abili

ty th

resh

old

(rpm

)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 11 The variation of thresholds of instability with ply anglefor various aspect ratios (flexural mode)

may be noted that the threshold of instability is larger thanthe critical rotating speed and the difference between themincreases as aspect ratio decreasesThis implies that the onsetof instability always occurs after the critical rotating speed

Figures 12 and 13 show the variation of the critical rotatingspeed and threshold of instability for the extension-twistmode respectively From these figures it becomes apparentthat the maximum ones occur at 120579 = 45

∘

4 Conclusion

A model was presented for the study of the dynamicalbehavior of rotating thin-walled composite shaft with inter-nal damping The presented model was used to predict

Shock and Vibration 9

600

550

500

450

350

400

300

250

200

0 10 20 30 40 50 60 70 80 90

Firs

t crit

ical

spee

d (r

pm)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 12The variation of critical speeds with ply angle for variousaspect ratios (extension-twist mode)

600

550

500

450

350

400

300

250

200

0 10 20 30 40 50 60 70 80 90

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Inst

abili

ty th

resh

old

(rpm

)

Figure 13 The variation of thresholds of instability with ply anglefor various aspect ratios (extension-twist mode)

the natural frequencies critical rotating speeds and insta-bility thresholds Theoretical solutions of the free vibrationof the shaft were determined by applying Galerkinrsquos methodFrom the present analysis and the numerical results thefollowing main conclusions were drawn

(1) The developed model provides means of predictingthe natural frequencies critical rotating speeds andinstability thresholds of rotating composite thin-walled shafts with internal damping

(2) The ply angle and aspect ratio affect the vibrationaland instability behavior of shaft significantly

(3) There is an obvious increase in the critical rotatingspeeds and instability thresholds as aspect ratio isdecreased

(4) For the flexural mode critical rotating speed andthreshold of instability have their maximum valuesat 120579 = 0

∘ while for the extension-twist mode themaximum ones occur at 120579 = 45

∘(5) The onset of instability always occurs after the critical

rotating speed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded by the National Natural Sci-ence Foundation of China (Grant no 11272190) ShandongProvincial Natural Science Foundation of China (Grant noZR2011EEM031) and Graduate Innovation Project of Shan-dong University of Science ampTechnology of China (Grant noYC130210)

References

[1] H Zinberg and M F Symonds ldquoThe development of anadvanced composite tail rotor drive shaftrdquo in Proceedings of the26th Annual National Forum of the American Helicopter SocietyWashington DC USA June 1970

[2] H L M dos Reis R B Goldman and P H Verstrate ldquoThin-walled laminated composite cylindrical tubes part III criticalspeed analysisrdquo Journal of Composites Technology and Researchvol 9 no 2 pp 58ndash62 1987

[3] C Kim and C W Bert ldquoCritical speed analysis of laminatedcomposite hollow drive shaftsrdquo Composites Engineering vol 3no 7-8 pp 633ndash643 1993

[4] S P Singh and K Gupta ldquoComposite shaft rotordynamic anal-ysis using a layerwise theoryrdquo Journal of Sound and Vibrationvol 191 no 5 pp 739ndash756 1996

[5] M Y Chang J K Chen and C Y Chang ldquoA simple spinninglaminated composite shaft modelrdquo International Journal ofSolids and Structures vol 41 no 3-4 pp 637ndash662 2004

[6] H B H Gubran and K Gupta ldquoThe effect of stacking sequenceand coupling mechanisms on the natural frequencies of com-posite shaftsrdquo Journal of Sound and Vibration vol 282 no 1-2pp 231ndash248 2005

[7] O Song N Jeong and L Librescu ldquoImplication of conservativeand gyroscopic forces on vibration and stability of an elasticallytailored rotating shaft modeled as a composite thin-walledbeamrdquo Journal of the Acoustical Society of America vol 109 no3 pp 972ndash981 2001

[8] L W Rehfield ldquoDesign analysis methodology for compositerotor bladesrdquo inProceedings of the 7thDoDNASAConference onFibrous Composites in Structural Design AFWAL-TR-85-3094pp V(a)1ndashV(a)15 Denver Colo USA 1985

[9] Y S Ren Q Y Dai and X Q Zhang ldquoModeling and dynamicanalysis of rotating composite shaftrdquo Journal of Vibroengineer-ing vol 15 no 4 pp 1816ndash1832 2013

[10] H L Wettergren and K O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous damping sup-ported in anisotropic bearingsrdquo Journal of Sound and Vibrationvol 195 no 1 pp 75ndash84 1996

10 Shock and Vibration

[11] S P Singh and K Gupta ldquoFree damped flexural vibrationanalysis of composite cylindrical tubes using beam and shelltheoriesrdquo Journal of Sound and Vibration vol 172 no 2 pp 171ndash190 1994

[12] A JMazzei andRA Scott ldquoEffects of internal viscous dampingon the stability of a rotating shaft driven through a universaljointrdquo Journal of Sound and Vibration vol 265 no 4 pp 863ndash885 2003

[13] O Montagnier and C Hochard ldquoDynamic instability of super-critical driveshafts mounted on dissipative supports-effects ofviscous and hysteretic internal dampingrdquo Journal of Sound andVibration vol 305 no 3 pp 378ndash400 2007

[14] W Kim A Argento and R A Scott ldquoForced vibration anddynamic stability of a rotating tapered composite Timoshenkoshaft bending motions in end-milling operationsrdquo Journal ofSound and Vibration vol 246 no 4 pp 583ndash600 2001

[15] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquo Composites Science andTechnology vol 68 no 2 pp 337ndash345 2008

[16] V Berdichevsky E Armanios and A Badir ldquoTheory ofanisotropic thin-walled closed-cross-section beamsrdquo Compos-ites Engineering vol 2 no 5ndash7 pp 411ndash432 1992

[17] Y S Ren X H Du S S Sun and X M Teng ldquoStructuraldamping of thin-walled composite one-cell beamsrdquo Journal ofVibration and Shock vol 31 no 3 pp 141ndash152 2012

[18] D A Saravanos D Varelis T S Plagianakos and N Chryso-choidis ldquoA shear beam finite element for the damping analysisof tubular laminated composite beamsrdquo Journal of Sound andVibration vol 291 no 3-5 pp 802ndash823 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

0 100 200 300 400 500 600 700

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

250

200

150

100

50

0

120579 = 0∘

120579 = 45∘

120579 = 90∘

120579 = 0∘

120579 = 45∘

120579 = 90∘

Figure 4 The variation of natural frequencies with the rotatingspeed for various ply angles (119886119887 = 36 first two extension-twistmodes)

600

400

200

0

minus200

minus400

minus600

minus8000 100 200 300 400 500 600 700

Dam

ping

(1s

)

Rotating speed Ω (rpm)

120579 = 0∘

120579 = 45∘

120579 = 90∘

Figure 5 The variation of dampings with the rotating speed forvarious ply angles (119886119887 = 36 first two extension-twist modes)

different from the case of flexural mode The maximumcritical speed can be reached when the ply angle 120579 = 45

∘Figure 5 shows the variation of the first two extension-

twist dampings versus rotating speed for various ply anglesIt can be observed that the second extension-twist mode isstable whereas the first extension-twist mode is unstable asthe rotating speed increases above certain value The self-excited range is easily identified from the figure by the signof damping It may also be noted that the effect of ply angleon the threshold of instability is similar to that previouslydescribed for the flexural mode

ab = 12

ab = 36

ab = 72

ab = 12

ab = 36

ab = 72

45

35

40

30

25

20

15

10

5

00 50 100 150 200

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

Figure 6 The variation of natural frequencies with the rotatingspeed for various aspect ratios (120579 = 45

∘ first two flexural modes)

60

40

20

0

minus20

minus40

minus60

minus80

minus100

minus120

minus1400 50 100 150 200

ab = 12

ab = 36

ab = 72

Dam

ping

(1s

)

Rotating speed Ω (rpm)

Figure 7 The variation of dampings with the rotating speed forvarious aspect ratios (120579 = 45

∘ first two flexural modes)

Figure 6 shows the variation of the first two flexuralnatural frequencies versus rotating speed for various aspectratios The results show that the critical rotating speedincreases with the decrease of aspect ratios

Figure 7 shows the variation of the first two flexuraldampings versus rotating speed for various aspect ratiosFrom the results it can be seen that the threshold of instabilityincreases as aspect ratio decreases

Figures 8 and 9 present the effect of aspect ratio onthe natural frequency-rotating speed curves and damping-rotating speed curves for the extension-twist mode respec-tivelyThe results show that the decrease of aspect ratio yields

8 Shock and Vibration

140

120

100

80

60

40

20

00 100 200 300 400 500 600 700

ab = 12

ab = 36

ab = 72

ab = 12

ab = 36

ab = 72

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

Figure 8 The variation of natural frequencies with the rotatingspeed for various aspect ratios (120579 = 45

∘ first two extension-twistmodes)

800

600

400

200

0

minus200

minus400

minus600

minus8000 100 200 300 400 500 600 700

Dam

ping

(1s

)

Rotating speed Ω (rpm)

ab = 12

ab = 36

ab = 72

Figure 9 The variation of dampings with the rotating speed forvarious aspect ratios (120579 = 45

∘ first two extension-twist modes)

a significant increase of the critical rotating speed and thethreshold of instability

Figure 10 shows the effect of ply angle on the criticalrotating speed for the flexural mode It can be seen that asthe ply angle increases the critical rotating speeds decreaseand the maximum critical speed is maximum at 120579 = 0

∘Figure 11 shows the effect of ply angle on the threshold of

instability for the flexural mode It is evident that the generaleffect of the ply angle and aspect ratio on the thresholdof instability is similar to that associated with the criticalrotating speeds By comparing Figure 10 with Figure 11 it

80

70

60

50

40

30

20

10

00

10 20 30 40 50 60 70 80 90

Firs

t crit

ical

spee

d (r

pm)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 10The variation of critical speeds with ply angle for variousaspect ratios (flexural mode)

80

70

60

50

40

30

20

10

00

10 20 30 40 50 60 70 80 90

Inst

abili

ty th

resh

old

(rpm

)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 11 The variation of thresholds of instability with ply anglefor various aspect ratios (flexural mode)

may be noted that the threshold of instability is larger thanthe critical rotating speed and the difference between themincreases as aspect ratio decreasesThis implies that the onsetof instability always occurs after the critical rotating speed

Figures 12 and 13 show the variation of the critical rotatingspeed and threshold of instability for the extension-twistmode respectively From these figures it becomes apparentthat the maximum ones occur at 120579 = 45

∘

4 Conclusion

A model was presented for the study of the dynamicalbehavior of rotating thin-walled composite shaft with inter-nal damping The presented model was used to predict

Shock and Vibration 9

600

550

500

450

350

400

300

250

200

0 10 20 30 40 50 60 70 80 90

Firs

t crit

ical

spee

d (r

pm)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 12The variation of critical speeds with ply angle for variousaspect ratios (extension-twist mode)

600

550

500

450

350

400

300

250

200

0 10 20 30 40 50 60 70 80 90

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Inst

abili

ty th

resh

old

(rpm

)

Figure 13 The variation of thresholds of instability with ply anglefor various aspect ratios (extension-twist mode)

the natural frequencies critical rotating speeds and insta-bility thresholds Theoretical solutions of the free vibrationof the shaft were determined by applying Galerkinrsquos methodFrom the present analysis and the numerical results thefollowing main conclusions were drawn

(1) The developed model provides means of predictingthe natural frequencies critical rotating speeds andinstability thresholds of rotating composite thin-walled shafts with internal damping

(2) The ply angle and aspect ratio affect the vibrationaland instability behavior of shaft significantly

(3) There is an obvious increase in the critical rotatingspeeds and instability thresholds as aspect ratio isdecreased

(4) For the flexural mode critical rotating speed andthreshold of instability have their maximum valuesat 120579 = 0

∘ while for the extension-twist mode themaximum ones occur at 120579 = 45

∘(5) The onset of instability always occurs after the critical

rotating speed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded by the National Natural Sci-ence Foundation of China (Grant no 11272190) ShandongProvincial Natural Science Foundation of China (Grant noZR2011EEM031) and Graduate Innovation Project of Shan-dong University of Science ampTechnology of China (Grant noYC130210)

References

[1] H Zinberg and M F Symonds ldquoThe development of anadvanced composite tail rotor drive shaftrdquo in Proceedings of the26th Annual National Forum of the American Helicopter SocietyWashington DC USA June 1970

[2] H L M dos Reis R B Goldman and P H Verstrate ldquoThin-walled laminated composite cylindrical tubes part III criticalspeed analysisrdquo Journal of Composites Technology and Researchvol 9 no 2 pp 58ndash62 1987

[3] C Kim and C W Bert ldquoCritical speed analysis of laminatedcomposite hollow drive shaftsrdquo Composites Engineering vol 3no 7-8 pp 633ndash643 1993

[4] S P Singh and K Gupta ldquoComposite shaft rotordynamic anal-ysis using a layerwise theoryrdquo Journal of Sound and Vibrationvol 191 no 5 pp 739ndash756 1996

[5] M Y Chang J K Chen and C Y Chang ldquoA simple spinninglaminated composite shaft modelrdquo International Journal ofSolids and Structures vol 41 no 3-4 pp 637ndash662 2004

[6] H B H Gubran and K Gupta ldquoThe effect of stacking sequenceand coupling mechanisms on the natural frequencies of com-posite shaftsrdquo Journal of Sound and Vibration vol 282 no 1-2pp 231ndash248 2005

[7] O Song N Jeong and L Librescu ldquoImplication of conservativeand gyroscopic forces on vibration and stability of an elasticallytailored rotating shaft modeled as a composite thin-walledbeamrdquo Journal of the Acoustical Society of America vol 109 no3 pp 972ndash981 2001

[8] L W Rehfield ldquoDesign analysis methodology for compositerotor bladesrdquo inProceedings of the 7thDoDNASAConference onFibrous Composites in Structural Design AFWAL-TR-85-3094pp V(a)1ndashV(a)15 Denver Colo USA 1985

[9] Y S Ren Q Y Dai and X Q Zhang ldquoModeling and dynamicanalysis of rotating composite shaftrdquo Journal of Vibroengineer-ing vol 15 no 4 pp 1816ndash1832 2013

[10] H L Wettergren and K O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous damping sup-ported in anisotropic bearingsrdquo Journal of Sound and Vibrationvol 195 no 1 pp 75ndash84 1996

10 Shock and Vibration

[11] S P Singh and K Gupta ldquoFree damped flexural vibrationanalysis of composite cylindrical tubes using beam and shelltheoriesrdquo Journal of Sound and Vibration vol 172 no 2 pp 171ndash190 1994

[12] A JMazzei andRA Scott ldquoEffects of internal viscous dampingon the stability of a rotating shaft driven through a universaljointrdquo Journal of Sound and Vibration vol 265 no 4 pp 863ndash885 2003

[13] O Montagnier and C Hochard ldquoDynamic instability of super-critical driveshafts mounted on dissipative supports-effects ofviscous and hysteretic internal dampingrdquo Journal of Sound andVibration vol 305 no 3 pp 378ndash400 2007

[14] W Kim A Argento and R A Scott ldquoForced vibration anddynamic stability of a rotating tapered composite Timoshenkoshaft bending motions in end-milling operationsrdquo Journal ofSound and Vibration vol 246 no 4 pp 583ndash600 2001

[15] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquo Composites Science andTechnology vol 68 no 2 pp 337ndash345 2008

[16] V Berdichevsky E Armanios and A Badir ldquoTheory ofanisotropic thin-walled closed-cross-section beamsrdquo Compos-ites Engineering vol 2 no 5ndash7 pp 411ndash432 1992

[17] Y S Ren X H Du S S Sun and X M Teng ldquoStructuraldamping of thin-walled composite one-cell beamsrdquo Journal ofVibration and Shock vol 31 no 3 pp 141ndash152 2012

[18] D A Saravanos D Varelis T S Plagianakos and N Chryso-choidis ldquoA shear beam finite element for the damping analysisof tubular laminated composite beamsrdquo Journal of Sound andVibration vol 291 no 3-5 pp 802ndash823 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

140

120

100

80

60

40

20

00 100 200 300 400 500 600 700

ab = 12

ab = 36

ab = 72

ab = 12

ab = 36

ab = 72

Rotating speed Ω (rpm)

Freq

uenc

y (H

z)

Figure 8 The variation of natural frequencies with the rotatingspeed for various aspect ratios (120579 = 45

∘ first two extension-twistmodes)

800

600

400

200

0

minus200

minus400

minus600

minus8000 100 200 300 400 500 600 700

Dam

ping

(1s

)

Rotating speed Ω (rpm)

ab = 12

ab = 36

ab = 72

Figure 9 The variation of dampings with the rotating speed forvarious aspect ratios (120579 = 45

∘ first two extension-twist modes)

a significant increase of the critical rotating speed and thethreshold of instability

Figure 10 shows the effect of ply angle on the criticalrotating speed for the flexural mode It can be seen that asthe ply angle increases the critical rotating speeds decreaseand the maximum critical speed is maximum at 120579 = 0

∘Figure 11 shows the effect of ply angle on the threshold of

instability for the flexural mode It is evident that the generaleffect of the ply angle and aspect ratio on the thresholdof instability is similar to that associated with the criticalrotating speeds By comparing Figure 10 with Figure 11 it

80

70

60

50

40

30

20

10

00

10 20 30 40 50 60 70 80 90

Firs

t crit

ical

spee

d (r

pm)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 10The variation of critical speeds with ply angle for variousaspect ratios (flexural mode)

80

70

60

50

40

30

20

10

00

10 20 30 40 50 60 70 80 90

Inst

abili

ty th

resh

old

(rpm

)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 11 The variation of thresholds of instability with ply anglefor various aspect ratios (flexural mode)

may be noted that the threshold of instability is larger thanthe critical rotating speed and the difference between themincreases as aspect ratio decreasesThis implies that the onsetof instability always occurs after the critical rotating speed

Figures 12 and 13 show the variation of the critical rotatingspeed and threshold of instability for the extension-twistmode respectively From these figures it becomes apparentthat the maximum ones occur at 120579 = 45

∘

4 Conclusion

A model was presented for the study of the dynamicalbehavior of rotating thin-walled composite shaft with inter-nal damping The presented model was used to predict

Shock and Vibration 9

600

550

500

450

350

400

300

250

200

0 10 20 30 40 50 60 70 80 90

Firs

t crit

ical

spee

d (r

pm)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 12The variation of critical speeds with ply angle for variousaspect ratios (extension-twist mode)

600

550

500

450

350

400

300

250

200

0 10 20 30 40 50 60 70 80 90

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Inst

abili

ty th

resh

old

(rpm

)

Figure 13 The variation of thresholds of instability with ply anglefor various aspect ratios (extension-twist mode)

the natural frequencies critical rotating speeds and insta-bility thresholds Theoretical solutions of the free vibrationof the shaft were determined by applying Galerkinrsquos methodFrom the present analysis and the numerical results thefollowing main conclusions were drawn

(1) The developed model provides means of predictingthe natural frequencies critical rotating speeds andinstability thresholds of rotating composite thin-walled shafts with internal damping

(2) The ply angle and aspect ratio affect the vibrationaland instability behavior of shaft significantly

(3) There is an obvious increase in the critical rotatingspeeds and instability thresholds as aspect ratio isdecreased

(4) For the flexural mode critical rotating speed andthreshold of instability have their maximum valuesat 120579 = 0

∘ while for the extension-twist mode themaximum ones occur at 120579 = 45

∘(5) The onset of instability always occurs after the critical

rotating speed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded by the National Natural Sci-ence Foundation of China (Grant no 11272190) ShandongProvincial Natural Science Foundation of China (Grant noZR2011EEM031) and Graduate Innovation Project of Shan-dong University of Science ampTechnology of China (Grant noYC130210)

References

[1] H Zinberg and M F Symonds ldquoThe development of anadvanced composite tail rotor drive shaftrdquo in Proceedings of the26th Annual National Forum of the American Helicopter SocietyWashington DC USA June 1970

[2] H L M dos Reis R B Goldman and P H Verstrate ldquoThin-walled laminated composite cylindrical tubes part III criticalspeed analysisrdquo Journal of Composites Technology and Researchvol 9 no 2 pp 58ndash62 1987

[3] C Kim and C W Bert ldquoCritical speed analysis of laminatedcomposite hollow drive shaftsrdquo Composites Engineering vol 3no 7-8 pp 633ndash643 1993

[4] S P Singh and K Gupta ldquoComposite shaft rotordynamic anal-ysis using a layerwise theoryrdquo Journal of Sound and Vibrationvol 191 no 5 pp 739ndash756 1996

[5] M Y Chang J K Chen and C Y Chang ldquoA simple spinninglaminated composite shaft modelrdquo International Journal ofSolids and Structures vol 41 no 3-4 pp 637ndash662 2004

[6] H B H Gubran and K Gupta ldquoThe effect of stacking sequenceand coupling mechanisms on the natural frequencies of com-posite shaftsrdquo Journal of Sound and Vibration vol 282 no 1-2pp 231ndash248 2005

[7] O Song N Jeong and L Librescu ldquoImplication of conservativeand gyroscopic forces on vibration and stability of an elasticallytailored rotating shaft modeled as a composite thin-walledbeamrdquo Journal of the Acoustical Society of America vol 109 no3 pp 972ndash981 2001

[8] L W Rehfield ldquoDesign analysis methodology for compositerotor bladesrdquo inProceedings of the 7thDoDNASAConference onFibrous Composites in Structural Design AFWAL-TR-85-3094pp V(a)1ndashV(a)15 Denver Colo USA 1985

[9] Y S Ren Q Y Dai and X Q Zhang ldquoModeling and dynamicanalysis of rotating composite shaftrdquo Journal of Vibroengineer-ing vol 15 no 4 pp 1816ndash1832 2013

[10] H L Wettergren and K O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous damping sup-ported in anisotropic bearingsrdquo Journal of Sound and Vibrationvol 195 no 1 pp 75ndash84 1996

10 Shock and Vibration

[11] S P Singh and K Gupta ldquoFree damped flexural vibrationanalysis of composite cylindrical tubes using beam and shelltheoriesrdquo Journal of Sound and Vibration vol 172 no 2 pp 171ndash190 1994

[12] A JMazzei andRA Scott ldquoEffects of internal viscous dampingon the stability of a rotating shaft driven through a universaljointrdquo Journal of Sound and Vibration vol 265 no 4 pp 863ndash885 2003

[13] O Montagnier and C Hochard ldquoDynamic instability of super-critical driveshafts mounted on dissipative supports-effects ofviscous and hysteretic internal dampingrdquo Journal of Sound andVibration vol 305 no 3 pp 378ndash400 2007

[14] W Kim A Argento and R A Scott ldquoForced vibration anddynamic stability of a rotating tapered composite Timoshenkoshaft bending motions in end-milling operationsrdquo Journal ofSound and Vibration vol 246 no 4 pp 583ndash600 2001

[15] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquo Composites Science andTechnology vol 68 no 2 pp 337ndash345 2008

[16] V Berdichevsky E Armanios and A Badir ldquoTheory ofanisotropic thin-walled closed-cross-section beamsrdquo Compos-ites Engineering vol 2 no 5ndash7 pp 411ndash432 1992

[17] Y S Ren X H Du S S Sun and X M Teng ldquoStructuraldamping of thin-walled composite one-cell beamsrdquo Journal ofVibration and Shock vol 31 no 3 pp 141ndash152 2012

[18] D A Saravanos D Varelis T S Plagianakos and N Chryso-choidis ldquoA shear beam finite element for the damping analysisof tubular laminated composite beamsrdquo Journal of Sound andVibration vol 291 no 3-5 pp 802ndash823 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

600

550

500

450

350

400

300

250

200

0 10 20 30 40 50 60 70 80 90

Firs

t crit

ical

spee

d (r

pm)

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Figure 12The variation of critical speeds with ply angle for variousaspect ratios (extension-twist mode)

600

550

500

450

350

400

300

250

200

0 10 20 30 40 50 60 70 80 90

Ply angle 120579 (deg)

ab = 12

ab = 18

ab = 36

ab = 72

Inst

abili

ty th

resh

old

(rpm

)

Figure 13 The variation of thresholds of instability with ply anglefor various aspect ratios (extension-twist mode)

the natural frequencies critical rotating speeds and insta-bility thresholds Theoretical solutions of the free vibrationof the shaft were determined by applying Galerkinrsquos methodFrom the present analysis and the numerical results thefollowing main conclusions were drawn

(1) The developed model provides means of predictingthe natural frequencies critical rotating speeds andinstability thresholds of rotating composite thin-walled shafts with internal damping

(2) The ply angle and aspect ratio affect the vibrationaland instability behavior of shaft significantly

(3) There is an obvious increase in the critical rotatingspeeds and instability thresholds as aspect ratio isdecreased

(4) For the flexural mode critical rotating speed andthreshold of instability have their maximum valuesat 120579 = 0

∘ while for the extension-twist mode themaximum ones occur at 120579 = 45

∘(5) The onset of instability always occurs after the critical

rotating speed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is funded by the National Natural Sci-ence Foundation of China (Grant no 11272190) ShandongProvincial Natural Science Foundation of China (Grant noZR2011EEM031) and Graduate Innovation Project of Shan-dong University of Science ampTechnology of China (Grant noYC130210)

References

[1] H Zinberg and M F Symonds ldquoThe development of anadvanced composite tail rotor drive shaftrdquo in Proceedings of the26th Annual National Forum of the American Helicopter SocietyWashington DC USA June 1970

[2] H L M dos Reis R B Goldman and P H Verstrate ldquoThin-walled laminated composite cylindrical tubes part III criticalspeed analysisrdquo Journal of Composites Technology and Researchvol 9 no 2 pp 58ndash62 1987

[3] C Kim and C W Bert ldquoCritical speed analysis of laminatedcomposite hollow drive shaftsrdquo Composites Engineering vol 3no 7-8 pp 633ndash643 1993

[4] S P Singh and K Gupta ldquoComposite shaft rotordynamic anal-ysis using a layerwise theoryrdquo Journal of Sound and Vibrationvol 191 no 5 pp 739ndash756 1996

[5] M Y Chang J K Chen and C Y Chang ldquoA simple spinninglaminated composite shaft modelrdquo International Journal ofSolids and Structures vol 41 no 3-4 pp 637ndash662 2004

[6] H B H Gubran and K Gupta ldquoThe effect of stacking sequenceand coupling mechanisms on the natural frequencies of com-posite shaftsrdquo Journal of Sound and Vibration vol 282 no 1-2pp 231ndash248 2005

[7] O Song N Jeong and L Librescu ldquoImplication of conservativeand gyroscopic forces on vibration and stability of an elasticallytailored rotating shaft modeled as a composite thin-walledbeamrdquo Journal of the Acoustical Society of America vol 109 no3 pp 972ndash981 2001

[8] L W Rehfield ldquoDesign analysis methodology for compositerotor bladesrdquo inProceedings of the 7thDoDNASAConference onFibrous Composites in Structural Design AFWAL-TR-85-3094pp V(a)1ndashV(a)15 Denver Colo USA 1985

[9] Y S Ren Q Y Dai and X Q Zhang ldquoModeling and dynamicanalysis of rotating composite shaftrdquo Journal of Vibroengineer-ing vol 15 no 4 pp 1816ndash1832 2013

[10] H L Wettergren and K O Olsson ldquoDynamic instability of arotating asymmetric shaft with internal viscous damping sup-ported in anisotropic bearingsrdquo Journal of Sound and Vibrationvol 195 no 1 pp 75ndash84 1996

10 Shock and Vibration

[11] S P Singh and K Gupta ldquoFree damped flexural vibrationanalysis of composite cylindrical tubes using beam and shelltheoriesrdquo Journal of Sound and Vibration vol 172 no 2 pp 171ndash190 1994

[12] A JMazzei andRA Scott ldquoEffects of internal viscous dampingon the stability of a rotating shaft driven through a universaljointrdquo Journal of Sound and Vibration vol 265 no 4 pp 863ndash885 2003

[13] O Montagnier and C Hochard ldquoDynamic instability of super-critical driveshafts mounted on dissipative supports-effects ofviscous and hysteretic internal dampingrdquo Journal of Sound andVibration vol 305 no 3 pp 378ndash400 2007

[14] W Kim A Argento and R A Scott ldquoForced vibration anddynamic stability of a rotating tapered composite Timoshenkoshaft bending motions in end-milling operationsrdquo Journal ofSound and Vibration vol 246 no 4 pp 583ndash600 2001

[15] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquo Composites Science andTechnology vol 68 no 2 pp 337ndash345 2008

[16] V Berdichevsky E Armanios and A Badir ldquoTheory ofanisotropic thin-walled closed-cross-section beamsrdquo Compos-ites Engineering vol 2 no 5ndash7 pp 411ndash432 1992

[17] Y S Ren X H Du S S Sun and X M Teng ldquoStructuraldamping of thin-walled composite one-cell beamsrdquo Journal ofVibration and Shock vol 31 no 3 pp 141ndash152 2012

[18] D A Saravanos D Varelis T S Plagianakos and N Chryso-choidis ldquoA shear beam finite element for the damping analysisof tubular laminated composite beamsrdquo Journal of Sound andVibration vol 291 no 3-5 pp 802ndash823 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

[11] S P Singh and K Gupta ldquoFree damped flexural vibrationanalysis of composite cylindrical tubes using beam and shelltheoriesrdquo Journal of Sound and Vibration vol 172 no 2 pp 171ndash190 1994

[12] A JMazzei andRA Scott ldquoEffects of internal viscous dampingon the stability of a rotating shaft driven through a universaljointrdquo Journal of Sound and Vibration vol 265 no 4 pp 863ndash885 2003

[13] O Montagnier and C Hochard ldquoDynamic instability of super-critical driveshafts mounted on dissipative supports-effects ofviscous and hysteretic internal dampingrdquo Journal of Sound andVibration vol 305 no 3 pp 378ndash400 2007

[14] W Kim A Argento and R A Scott ldquoForced vibration anddynamic stability of a rotating tapered composite Timoshenkoshaft bending motions in end-milling operationsrdquo Journal ofSound and Vibration vol 246 no 4 pp 583ndash600 2001

[15] R Sino T N Baranger E Chatelet and G Jacquet ldquoDynamicanalysis of a rotating composite shaftrdquo Composites Science andTechnology vol 68 no 2 pp 337ndash345 2008

[16] V Berdichevsky E Armanios and A Badir ldquoTheory ofanisotropic thin-walled closed-cross-section beamsrdquo Compos-ites Engineering vol 2 no 5ndash7 pp 411ndash432 1992

[17] Y S Ren X H Du S S Sun and X M Teng ldquoStructuraldamping of thin-walled composite one-cell beamsrdquo Journal ofVibration and Shock vol 31 no 3 pp 141ndash152 2012

[18] D A Saravanos D Varelis T S Plagianakos and N Chryso-choidis ldquoA shear beam finite element for the damping analysisof tubular laminated composite beamsrdquo Journal of Sound andVibration vol 291 no 3-5 pp 802ndash823 2006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of